  We introduce the notion of a \emph{locally scaling} transformation defined on
a compact-open subset of a non-archimedean local field. We show that this class
encompasses the Haar measure-preserving transformations defined by ${\mathcal
C}^1$ (in particular, polynomial) maps, and prove a structure theorem for
locally scaling transformations. We use the theory of polynomial approximation
on compact-open subsets of non-archimedean local fields to demonstrate the
existence of ergodic Markov, and mixing Markov transformations defined by such
polynomial maps. We also give simple sufficient conditions on the Mahler
expansion of a continuous map $\mathbb Z_p \to \mathbb Z_p$ for it to define a
Bernoulli transformation.
