Automorphic Numbers with 5 Digits

Theorem
The only $5$-digit automorphic number which does not begin with a zero is $90 \, 625$.

Proof
We have:
 * $90 \, 625^2 = 8 \, 212 \, 8 \mathbf {90 \, 625}$

thus demonstrating it is automorphic.

By Automorphic Numbers in Base 10, the only other possible candidate is $6^{5^4}$.

However:
 * $6^{5^4} \equiv 09 \, 376 \pmod {10^5}$

begins with a zero.

Hence there are no others.