A significant step in the Pila–Zannier approach to the André–Oort conjecture is a geometric transcendence result for the uniformization map of modular curves. I will discuss joint work with François Loeser. We prove an analogue of this result in non-archimedean geometry, namely for the uniformization of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of PGL(2,ℚₚ) contained in PGL(2,ℚ). In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.