In Gouvea book $p$-adic numbers, on can find this corollary (7.4.11)
Let $f(X) = 1+a_1X+a_2X^2+a_3X^3+\cdots$ be a power series which converges on the closed ball of radius $c = p^m$. Let $m_1, m_2, \cdots, m_k$ be the slopes of the Newton polygon of $f(X)$ which are less than or equal to $m$, and let $i_1, i_2,\cdots , i_k$ be their lengths. Then, for each $j$, $f(X)$ has $i_j$ zeros with absolute value $p^{m_j}$ , and there are no other zeros in the closed ball of radius $p^m$.
This is in $\mathbb C_p$. Does such a theorem exist for any non-archimedean complete algebraically closed field $K$, specially when $K$ is of characteristic $p>0$. In Gouvea, proof of the corollary uses Hensel lemma that, in turn, uses derivative. Since derivative can be zero in positive characteristic, it seems that Gouvea proof can not be exported as such in positive characteristic context.
