Let $\Gamma$ be an arithmetic lattice in a linear algebraic $\mathbb{Q}$-group $\mathbf{G}$, that is, $\Gamma$ is a subgroup of $\mathbf{G}(\mathbb{Q})$ that is commensurable with $\mathbf{G}(\mathbb{Z})$.

For a prime $p$, we can consider $\Gamma$ as a subspace of $\mathbf{G}(\mathbb{Q}_p)$. My question is:

What does the closure of $\Gamma$ in $\mathbf{G}(\mathbb{Q}_p)$ with respect to the $p$-adic topology look like?

The closure of $\mathbb{Z}$ in $\mathbb{Q}_p$ is the ring of $p$-adic integers $\mathbb{Z}_p$. So it seems plausible to me that, for example, the closure of the lattice $\Gamma = SL(n,\mathbb{Z})$ in the $p$-adic topology of $SL(n,\mathbb{Q}_p)$ would be $SL(n,\mathbb{Z}_p)$. Is this correct?

Also, what about other lattice, for example, what is the closure of a congruence subgroup $$\Gamma(c) := \{ g \in SL(n,\mathbb{Z}) : g - I_n \equiv 0 \;\text{ mod } c\, \} \subset SL(n,\mathbb{Z})$$ in the $p$-adic topology of $SL(n,\mathbb{Q}_p)$?