On a special congruence of Carlitz

We prove that if $q$ is a power of a prime $p$ and $p^k$ divides $a$, with $k\ge 0$, then $$1+(q-1)\sum_{0\le b(q-1)<a} \binom{a}{b(q-1)}\equiv 0\pmod{p^{k+1}}.$$ The special case of this congruence where $q=p$ was proved by Carlitz in 1953 by means of rather deep properties of the Bernoulli numbers. A more direct approach produces our generalization and several related results.