  We consider a family of polynomial analogs of the Ap\'ery numbers, which
includes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show that
the supercongruences that Gessel and Mimura established for the Ap\'ery numbers
generalize to these polynomials. Our proof relies on polynomial analogs of
classical binomial congruences of Wolstenholme and Ljunggren. We further
indicate that this approach generalizes to other supercongruence results.
