  In this work we study $p$-adic continuous functions in several variables
taking values on $\mathbb{Z}_p$. We describe the orthonormal van der Put base
of these functions and study various Lipschitz conditions in several variables,
generalizing previous work of Anashin. In particular, we introduce a new class
of $p$-adic Lipschitz functions and study some of their properties. We also
prove a Hensel's lifting lemma for this new class of functions, generalizing
previous results of Yurova and Khrennikov.
