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Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ fora fixed $p>2$. Projective general linear groups$G=\operatorname{PGL}_{2}(\mathbb{K})$ and$H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on Bruhat-Titstrees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of$H$-subtrees $G.T_{H}$. Let $\Gamma$ be a Schottky subgroup such that$\Gamma\backslash T_{G}$ is infinite volume and has an additional conditionnamed high-branchedness, and let $\Lambda$ be its limit set.  We classify $\Gamma$-orbits in $G/H$. Let $C=g_{C}H\in G/H$. As ageneralization of Ratner's theorem, if $\Gamma\backslash g_{C}.T_{H}$ meets theconvex core of $\Gamma\backslash T_{G}$, then the $\Gamma$-orbit of $C$ iseither dense or closed in $ {\cal{C}}_{\Lambda}=\{g H:\partial(g.T_{H})\cap\Lambda\neq\varnothing\}$.

$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-orbit closures on a $p$-adic homogeneous space of infinite volume