What are the examples of lattices in $SL_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?

It is know $SO_n(\mathbb{Z}[1/p])$ is a lattice in $SO_n(\mathbb{Q}_p)$ and, in general, $G(\mathbb{Z}[1/p_i,1\leq i\leq m])$ is a lattice in $G(\mathbb{R})\times \Pi_{1\leq i \leq m}G(\mathbb{Q}_{p_i})$. How can we exclude the archimedean places and consider a single prime $p$?

It is also known that irreducible lattice in $G$ are (S-)-arithmetic if the rank of $G$ is no less than 2. (See Margulis’ book ${\it Discrete~Subgroups~of~Semisimple~Lie~Groups}$). But it is always assumed $S$ contains the infinite places.

A similar question

What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?

It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a lattice in $\operatorname{SO}_n(\mathbb{Q}_p)$ and, in general, $G(\mathbb{Z}[1/p_i,1\leq i\leq m])$ is a lattice in $G(\mathbb{R})\times \Pi_{1\leq i \leq m}G(\mathbb{Q}_{p_i})$. How can we exclude the archimedean places and consider a single prime $p$?

It is also known that irreducible lattice in $G$ are (S-)-arithmetic if the rank of $G$ is no less than 2. (See Margulis’ book Discrete Subgroups of Semisimple Lie Groups). But it is always assumed $S$ contains the infinite places.

A similar question: discrete subgroups in p-adic Lie groups?.