  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 $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm
PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic
irreducible subvarieties of the uniformization.
