We shall discuss versions of the Yomdin-Gromov reparametrization lemma valid in a non-archimedean setting. Special care is needed in order to insure these parametrizations are globally well approximated by their Taylor expansions. We shall present as applications a $p$-adic version of the Pila-Wilkie theorem and a geometric analogue of estimates of Bombieri-Pila and Pila for the number of integral points of bounded height on affine algebraic varieties. This is joint work with R. Cluckers and G. Comte.