A non-archimedean Ax-Lindemann theorem

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $(2,\overline{{\mathbf{Q}}_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.