Compact quotients of non-Archimedean rank-one groups

Let G be the set of points of a connected semisimple algebraic group of relative rank one over a non-Archimedean local field. We describe all finitely generated torsion-free discrete subgroups of G x G acting properly discontinuously and cocompactly on G by left and right multiplication. We prove that after a small deformation in G x G such a subgroup keeps acting freely, properly discontinuously, and cocompactly on G.