We begin with a brute force search for permutable primes less than 10^6.
We only list the permutable primes whose digits are in ascending order.
The one-digit primes are omitted.

     11 13 17 37 79 113 199 337 

We will prove that no other permutable primes exist, except for repunits.
The proof uses a lemma: If N and m are integers greater than 1 such that
the digit permutations of N cover all congruence classes modulo m, then
no permutable prime can contain all of the digits of N (as a multiset).
By choosing a finite number of pairs (N,m) we can rule out all numbers
having six or more digits, except for repunits.

Case 1: N has four distinct digits (1,3,7,9). We use m = 7. 

    1379 = 0 (mod 7)
    1793 = 1 (mod 7)
    3719 = 2 (mod 7)
    1739 = 3 (mod 7)
    1397 = 4 (mod 7)
    1937 = 5 (mod 7)
    1973 = 6 (mod 7)

Case 2: N has three distinct digits.

    Case 2A: N contains aabbc.

        13713 = 0 (mod 7)
        11733 = 1 (mod 7)
        17313 = 2 (mod 7)
        13317 = 3 (mod 7)
        11337 = 4 (mod 7)
        11373 = 5 (mod 7)
        13173 = 6 (mod 7)

        13139 = 0 (mod 7)
        13931 = 1 (mod 7)
        19133 = 2 (mod 7)
        31139 = 3 (mod 7)
        11393 = 4 (mod 7)
        11933 = 5 (mod 7)
        11339 = 6 (mod 7)

        17731 = 0 (mod 7)
        17137 = 1 (mod 7)
        11377 = 2 (mod 7)
        17713 = 3 (mod 7)
        17371 = 4 (mod 7)
        11737 = 5 (mod 7)
        11773 = 6 (mod 7)

        11977 = 0 (mod 7)
        17179 = 1 (mod 7)
        11797 = 2 (mod 7)
        19771 = 3 (mod 7)
        17791 = 4 (mod 7)
        11779 = 5 (mod 7)
        71791 = 6 (mod 7)

        91931 = 0 (mod 7)
        19139 = 1 (mod 7)
        11993 = 2 (mod 7)
        11399 = 3 (mod 7)
        11939 = 4 (mod 7)
        19913 = 5 (mod 7)
        19193 = 6 (mod 7)

        19971 = 0 (mod 7)
        17991 = 1 (mod 7)
        11979 = 2 (mod 7)
        19197 = 3 (mod 7)
        11799 = 4 (mod 7)
        91971 = 5 (mod 7)
        11997 = 6 (mod 7)

        37317 = 0 (mod 7)
        37731 = 1 (mod 7)
        37137 = 2 (mod 7)
        33771 = 3 (mod 7)
        33177 = 4 (mod 7)
        33717 = 5 (mod 7)
        31737 = 6 (mod 7)

        37793 = 0 (mod 7)
        33797 = 1 (mod 7)
        37739 = 2 (mod 7)
        37397 = 3 (mod 7)
        33779 = 4 (mod 7)
        37973 = 5 (mod 7)
        33977 = 6 (mod 7)

        39319 = 0 (mod 7)
        93913 = 1 (mod 7)
        39391 = 2 (mod 7)
        39931 = 3 (mod 7)
        33919 = 4 (mod 7)
        33199 = 5 (mod 7)
        33991 = 6 (mod 7)

        39739 = 0 (mod 7)
        33979 = 1 (mod 7)
        39937 = 2 (mod 7)
        33799 = 3 (mod 7)
        39379 = 4 (mod 7)
        33997 = 5 (mod 7)
        37939 = 6 (mod 7)

        71799 = 0 (mod 7)
        97791 = 1 (mod 7)
        77919 = 2 (mod 7)
        77199 = 3 (mod 7)
        77991 = 4 (mod 7)
        79791 = 5 (mod 7)
        79197 = 6 (mod 7)

        77399 = 0 (mod 7)
        77939 = 1 (mod 7)
        79739 = 2 (mod 7)
        79397 = 3 (mod 7)
        79937 = 4 (mod 7)
        79973 = 5 (mod 7)
        77993 = 6 (mod 7)

    Case 2B: N contains aaaaabc.

        1111173 = 0 (mod 7)
        1111713 = 1 (mod 7)
        1111371 = 2 (mod 7)
        1113171 = 3 (mod 7)
        1111317 = 4 (mod 7)
        1111731 = 5 (mod 7)
        1111137 = 6 (mod 7)

        1113119 = 0 (mod 7)
        1111139 = 1 (mod 7)
        1111931 = 2 (mod 7)
        1131119 = 3 (mod 7)
        1119311 = 4 (mod 7)
        1111913 = 5 (mod 7)
        1111193 = 6 (mod 7)

        1111719 = 0 (mod 7)
        7111119 = 1 (mod 7)
        1111791 = 2 (mod 7)
        1111197 = 3 (mod 7)
        1117911 = 4 (mod 7)
        1117191 = 5 (mod 7)
        1111179 = 6 (mod 7)

        3337313 = 0 (mod 7)
        3333317 = 1 (mod 7)
        3333731 = 2 (mod 7)
        3333137 = 3 (mod 7)
        3333173 = 4 (mod 7)
        3333713 = 5 (mod 7)
        3333371 = 6 (mod 7)

        3339133 = 0 (mod 7)
        3313339 = 1 (mod 7)
        3333913 = 2 (mod 7)
        3333319 = 3 (mod 7)
        3331339 = 4 (mod 7)
        3333391 = 5 (mod 7)
        3333931 = 6 (mod 7)

        3333379 = 0 (mod 7)
        3333793 = 1 (mod 7)
        3339373 = 2 (mod 7)
        3333739 = 3 (mod 7)
        3333397 = 4 (mod 7)
        3333937 = 5 (mod 7)
        3333973 = 6 (mod 7)

        7777371 = 0 (mod 7)
        7771737 = 1 (mod 7)
        7777317 = 2 (mod 7)
        7777731 = 3 (mod 7)
        7777137 = 4 (mod 7)
        7777173 = 5 (mod 7)
        7777713 = 6 (mod 7)

        7777791 = 0 (mod 7)
        7777197 = 1 (mod 7)
        7719777 = 2 (mod 7)
        7771977 = 3 (mod 7)
        7777179 = 4 (mod 7)
        7777719 = 5 (mod 7)
        7779771 = 6 (mod 7)

        7777973 = 0 (mod 7)
        7777379 = 1 (mod 7)
        7777793 = 2 (mod 7)
        7773797 = 3 (mod 7)
        7777739 = 4 (mod 7)
        7777397 = 5 (mod 7)
        7777937 = 6 (mod 7)

        9999913 = 0 (mod 7)
        9999193 = 1 (mod 7)
        1999993 = 2 (mod 7)
        9999139 = 3 (mod 7)
        9999931 = 4 (mod 7)
        9991399 = 5 (mod 7)
        9991939 = 6 (mod 7)

        9917999 = 0 (mod 7)
        9999179 = 1 (mod 7)
        9999971 = 2 (mod 7)
        9997991 = 3 (mod 7)
        9999917 = 4 (mod 7)
        9999197 = 5 (mod 7)
        9991799 = 6 (mod 7)

        9997939 = 0 (mod 7)
        9999739 = 1 (mod 7)
        9999397 = 2 (mod 7)
        9999937 = 3 (mod 7)
        9999973 = 4 (mod 7)
        9999379 = 5 (mod 7)
        9999793 = 6 (mod 7)

Case 3: N has two distinct digits.

    Case 3A: N contains aaabb.

        31311 = 0 (mod 7)
        11313 = 1 (mod 7)
        13113 = 2 (mod 7)
        11133 = 3 (mod 7)
        13311 = 4 (mod 7)
        11331 = 5 (mod 7)
        13131 = 6 (mod 7)

        17171 = 0 (mod 7)
        17711 = 1 (mod 7)
        17117 = 2 (mod 7)
        71711 = 3 (mod 7)
        11771 = 4 (mod 7)
        11177 = 5 (mod 7)
        11717 = 6 (mod 7)

        11991 = 0 (mod 7)
        91911 = 1 (mod 7)
        19119 = 2 (mod 7)
        19911 = 3 (mod 7)
        19191 = 4 (mod 7)
        11919 = 5 (mod 7)
        11199 = 6 (mod 7)

        33131 = 0 (mod 7)
        13133 = 1 (mod 7)
        31313 = 2 (mod 7)
        33113 = 3 (mod 7)
        31133 = 4 (mod 7)
        33311 = 5 (mod 7)
        31331 = 6 (mod 7)

        37373 = 0 (mod 7)
        33377 = 1 (mod 7)
        73733 = 2 (mod 7)
        37733 = 3 (mod 7)
        33737 = 4 (mod 7)
        33773 = 5 (mod 7)
        37337 = 6 (mod 7)

        93933 = 0 (mod 7)
        33993 = 1 (mod 7)
        33399 = 2 (mod 7)
        33939 = 3 (mod 7)
        39393 = 4 (mod 7)
        39933 = 5 (mod 7)
        39339 = 6 (mod 7)

        71771 = 0 (mod 7)
        71177 = 1 (mod 7)
        71717 = 2 (mod 7)
        77171 = 3 (mod 7)
        77711 = 4 (mod 7)
        77117 = 5 (mod 7)
        17177 = 6 (mod 7)

        73773 = 0 (mod 7)
        77337 = 1 (mod 7)
        77373 = 2 (mod 7)
        73377 = 3 (mod 7)
        37377 = 4 (mod 7)
        77733 = 5 (mod 7)
        73737 = 6 (mod 7)

        79779 = 0 (mod 7)
        77799 = 1 (mod 7)
        79977 = 2 (mod 7)
        77997 = 3 (mod 7)
        79797 = 4 (mod 7)
        97977 = 5 (mod 7)
        77979 = 6 (mod 7)

        99911 = 0 (mod 7)
        99191 = 1 (mod 7)
        91919 = 2 (mod 7)
        91199 = 3 (mod 7)
        91991 = 4 (mod 7)
        19199 = 5 (mod 7)
        99119 = 6 (mod 7)

        99393 = 0 (mod 7)
        99933 = 1 (mod 7)
        99339 = 2 (mod 7)
        39399 = 3 (mod 7)
        93993 = 4 (mod 7)
        93399 = 5 (mod 7)
        93939 = 6 (mod 7)

        97979 = 0 (mod 7)
        99779 = 1 (mod 7)
        97799 = 2 (mod 7)
        99977 = 3 (mod 7)
        97997 = 4 (mod 7)
        99797 = 5 (mod 7)
        79799 = 6 (mod 7)

    Case 3B: N is a near-repdigit (one digit is different from the others).

         We show that a near-repunit cannot be prime if it has 17 or more digits.
         Suppose that N = aaa...ab, where a is repeated 16 or more times.
         Then the numbers N - (b-a) + (b-a)*10^k are permutations of N
         for k = 0, 1, 2, ..., 16. These numbers cover all congruence classes
         modulo 17, since 10 is a primitive root. Therefore, 17 divides one of
         these numbers.

         We check (using the Miller-Rabin primality test) that the following
         numbers are NOT prime.

            1111113
            1111117
            1111119
            3313333
            3333337
            3333339
            7777771
            7777773
            7777779
            9999919
            9999993
            9999997

            11111113
            11111711
            11111191
            33333313
            33333337
            33333339
            77777771
            77777773
            77777779
            99999991
            99999993
            99999997

            111111311
            111111117
            111111119
            333333331
            333333337
            333333339
            777777771
            777777737
            777777779
            999999991
            999999993
            999999997

            1111111113
            1111111117
            1111111119
            3333333331
            3333333337
            3333333339
            7777777771
            7777777773
            7777777779
            9999999991
            9999999993
            9999999997

            11111111311
            11111111117
            11111111119
            33333333331
            33333333337
            33333333339
            77777777771
            77777777773
            77777777779
            99999999991
            99999999993
            99999999997

            111111111113
            111111111117
            111111111119
            333333333331
            333333333337
            333333333339
            777777777771
            777777777737
            777777777779
            999999999991
            999999999993
            999999999997

            1111111111113
            1111111111117
            1111111111119
            3333333333331
            3333333333337
            3333333333339
            7777777777717
            7777777777773
            7777777777779
            9999999999991
            9999999999993
            9999999999997

            11111111111113
            11111111111117
            11111111111119
            33333333333331
            33333333333337
            33333333333339
            77777777777771
            77777777777773
            77777777777779
            99999999999991
            99999999999993
            99999999999997

            111111111111113
            111111111111117
            111111111111119
            333333333333331
            333333333333337
            333333333333339
            777777777777771
            777777777777737
            777777777777779
            999999999999991
            999999999999993
            999999999999997

            1111111111111113
            1111111111111117
            1111111111111119
            3333333333333331
            3333333333333337
            3333333333333339
            7777777777777771
            7777777777777773
            7777777777777779
            9999999999999991
            9999999999999993
            9999999999999997

QED.