This is a problem from the SUMS (Sydney University Mathematical Society) problems competition circa 1982 (open to undergraduates at all Australian universities and colleges). I couldn't prove it at the time, and having recently found it in a desk draw while having a clean up I still can't prove it.
I think it's relatively easy to show that if any such $m^n$ exists it must be of the form $5^n$. But it's not at all clear to me how to draw out a contradication to prove that no such $5^n$ exists for any $p$ which is prime. I suspect that Fermat's little theorem may play a role in there somewhere.
Many thanks for any guidance that you can offer.