  Let $\mathcal L_1$ be the set of all mappings $f\colon\Z_p\Z_p$ of the space
of all $p$-adic integers $\Z_p$ into itself that satisfy Lipschitz condition
with a constant 1. We prove that the mapping $f\in\mathcal L_1$ is ergodic with
respect to the normalized Haar measure on $\Z_p$ if and only if $f$ induces a
single cycle permutation on each residue ring $\Z/p^k\Z$ modulo $p^k$, for all
$k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings,
are considered also.
  Results of the paper in a combination with earlier results of the author give
explicit description of ergodic mappings from $\mathcal L_1$. This
characterization is complete for $p=2$.
  As an application we obtain a characterization of polynomials (and certain
locally analytic functions) that induce ergodic transformations of $p$-adic
spheres. The latter result implies a solution of a problem (posed by
A.~Khrennikov) about the ergodicity of a perturbed monomial mapping on a
sphere.
