In this chapter, we, following Anashin (Discrete Math Appl 12(6):527–590, 2002), introduce and examine three important special classes, \(\mathcal A\) , \(\mathcal B\) , and \(\mathcal C\) , of p-adic 1-Lipschitz functions. These functions play an important role in the p-adic ergodic theory (see Part II). All these functions belong to a wider class, the class of locally analytic functions. The class \(\mathcal C\) is the class of all power series over \(\mathbb Z_p\) , which converge everywhere in \(\mathbb Z_p\) ; the class \(\mathcal B\) is a completion of \(\mathcal C\) with respect to \(\sup \) -norm.