In this chapter, we find ergodicity and/or measure-preservation conditions for functions \(F\colon \mathbb {Z}_p^n\rightarrow \mathbb {Z}_p^m\) that are uniformly differentiable modulo p and that have integer-valued derivatives modulo p. Recall that in view of Theorem 5.5 , all these functions are locally 1-Lipschitz (or, in other terms, locally compatible), that is, these are the functions that are 1-Lipschitz on all sufficiently small balls. Therefore, for these functions F, the reduced maps \(F\mathbin {\mathsf {mod}} p^k\colon (\mathbb {Z}/p^k\mathbb {Z})^n\rightarrow (\mathbb {Z}/p^k\mathbb {Z})^m\) are well defined, provided k is sufficiently large, say, \(k\geqslant N_1(F)\) (cf. Definition 3.22 ). Thus, we can apply Theorem 8.2 to study measure-preservation and ergodicity of F. As 1-Lipschitz uniformly differentiable functions are a special case of functions under consideration, the theory that follows can be applied to various important classes of functions, e.g., for analytic functions on \(\mathbb {Z}_p\) , \(\mathcal {C}\) -functions, \(\mathcal {B}\) -functions, \(\mathcal {A}\) -functions (in particular, for twice integer-valued polynomials over \(\mathbb {Q}_p\) ), etc.