Strassman's Theorem

Let $(K, \vert\cdot\vert)$ be a complete non-Archimedean Valuated Field, with Valuation Ring $R$, and let $f(X)$ be a Power series with Coefficients in $R$. Suppose at least one of the Coefficients is Nonzero (so that $f$ is not identically zero) and the sequence of Coefficients converges to 0 with respect to $\vert\cdot\vert$. Then $f(X)$ has only finitely many zeros in $R$.

See also Archimedean Valuation, Mahler-Lech Theorem, Valuation, Valuation Ring