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We give lower bounds for the size of linearization discs for power seriesover $\mathbb{C}_p$. For quadratic maps, and certain power series containing a`sufficiently large' quadratic term, we find the exact linearization disc. Forfinite extensions of $\mathbb{Q}_p$, we give a sufficient condition on themultiplier under which the corresponding linearization disc is maximal (i.e.its radius coincides with that of the maximal disc in $\mathbb{C}_p$ on which$f$ is one-to-one). In particular, in unramified extensions of $\mathbb{Q}_p$,the linearization disc is maximal if the multiplier map has a maximal cycle onthe unit sphere. Estimates of linearization discs in the remaining types ofnon-Archimedean fields of dimension one were obtained in\cite{Lindahl:2004,Lindahl:2009,Lindahl:2009eq}.  Moreover, it is shown that, for any complete non-Archimedean field,transitivity is preserved under analytic conjugation. Using results by Oxtoby\cite{Oxtoby:1952}, we prove that transitivity, and hence minimality, isequivalent the unique ergodicity on compact subsets of a linearization disc. Inparticular, a power series $f$ over $\mathbb{Q}_p$ is minimal, hence uniquelyergodic, on all spheres inside a linearization disc about a fixed point if andonly if the multiplier is maximal. We also note that in finite extensions of$\mathbb{Q}_p$, as well as in any other non-Archimedean field $K$ that is notisomorphic to $\mathbb{Q}_p$ for some prime $p$, a power series cannot beergodic on an entire sphere, that is contained in a linearization disc, andcentered about the corresponding fixed point.

Estimates of linearization discs in $p$-adic dynamics with application to ergodicity