  A $p$-adic version of Gromov-Witten invariants for counting plane curves of
genus $g$ and degree $d$ through a given number of points is discussed. The
multiloop version of $p$-adic string theory considered by Chekhov and others
motivates us to ask how many of these curves are Mumford curves, i.e.\
uniformisable by a domain at the boundary of the Bruhat-Tits tree for
$\PGL_2(\mathbb{Q}_p)$. Generally, the number of Mumford curves depends on the
position of the given points in $\mathbb{P}^2$. With the help of tropical
geometry we find configurations of points through which all curves of given
degree and genus are Mumford curves. The article is preceded by an introduction
to some concepts of $p$-adic geometry and their relation to string theory.
