  Let $E$ be an elliptic curve over an algebraically closed, complete,
non-archimedean field $K$, and let ${\mathsf E}$ denote the Berkovich analytic
space associated to $E/K$. We study the $\mu$-equidistribution of finite
subsets of $E(K)$, where $\mu$ is a certain canonical unit Borel measure on
${\mathsf E}$. Our main result is an inequality bounding the error term when
testing against a certain class of continuous functions on ${\mathsf E}$. We
then give two applications to elliptic curves over global function fields: we
prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution
theorem for small points, and a function field analogue of a result of
Baker-Ih-Rumely on the finiteness of $S$-integral torsion points. Both
applications are given in explicit quantitative form.
