Ping-pong in the projective plane over a nonarchimedean field

We show that any lattice in ${\mathrm{S}\mathrm{L}}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product ${\mathbb{Z}}^2\ast \mathbb{Z}$. To our knowledge, the subgroups we construct give the first examples in the literature of finitely generated Zariski-dense infinite-covolume discrete subgroups of an almost simple group over a nonarchimedean local field that are not virtually free. Our result is in contrast to the case of ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$, in which the existence of a ${\mathbb{Z}}^2\ast \mathbb{Z}$ subgroup remains open.