Using Dwork's theory, the authors prove a broad generalization of his famous $p$-adic formal congruences theorem. This enables them to prove certain $p$-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number $p$ and not only for almost all primes. Furthermore, using Christol's functions, the authors provide an explicit formula for the &'grave;Eisenstein constant'' of any hypergeometric series with rational parameters. As an application of these results, the authors obtain an arithmetic statement &'grave;on average'' of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.