In this chapter, we address the ergodicity of 1-Lipschitz transformations on certain subspaces S of the space \(\mathbb {Z}_p\) , namely, on balls and on spheres of non-zero radii. The subspaces are endowed with a natural probability measure \(\hat \mu _p\) induced on S by the probability measure \(\mu _p\) on the whole space \(\mathbb {Z}_p\) ; i.e., \(\hat \mu _p(A)=\mu _p(A)/\mu _p(S)\) for each \(\mu _p\) -measurable subset \(A\subset S\) , so \(\hat \mu _p(S)=1\) . Now, if \(f\colon S\to S\) is a 1-Lipschitz map, we can speak of ergodicity of this map with respect to the measure \(\hat \mu _p\) . In the sequel, speaking of ergodicity (and of measure preservation) of a map f on a subspace S, we mean that S is invariant under action of f and the measure is \(\hat \mu _p\) .