Nonarchimedean discrete groups and quasi-isometric doubles of free groups

For any nonarchimedean local field $k$, we exhibit the first examples of Zariski-dense, discrete subgroups of special linear groups over $k$, which are not commensurable to any lattice in the points of a semisimple group over a local field, and contain an infinite index subgroup isomorphic to a lattice in the $k$-points of a higher rank $k$-semisimple group. We prove that any Gromov-hyperbolic double of a non-elementary free group along a cyclic subgroup embeds as a quasi-isometrically embedded subgroup of ${\mathsf{S}\mathsf{L}}_5(k^{\mathrm{\prime }})\times {\mathsf{S}\mathsf{L}}_5(k^{\mathrm{\prime }})$, where $k^{\mathrm{\prime }}$ is a discretely valued field. More precisely, $k^{\mathrm{\prime }}$ is the field of rational functions $k(t)$ equipped with the Gauss valuation and $k$ is any nonarchimedean local field of characteristic not equal to $2$ or $3$.