Chabauty limits of algebraic groups acting on trees

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\text{Aut} (T)$. We show that the set of closed cocompact subgroups of $\text{Aut} (T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\text{Aut} (T)$. This is done via a study of the integral Bruhat$-$Tits model of $\text{SL}_2$ and $\text{SU}_3^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\text{Aut} (T)$ is not always preserved under Chabauty limits.