Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially says that if $G$ is simple connected algebraic group over a local field $K$, any compact subgroup of $G(K)$ is open in an $H(E)$, where $E$ is a sub-local field of $K$ and $H$ a form of $G$ over $K$, and explains how to determine this $E$ and $H$. "Essentially" because the true statement of the theorem is complicated by several issues (adjoint vs simply connected, special isogenies, etc.). So let me state a special case where $E=K$ and $H=G$, which I believe (is correct and) gives an idea of the general statement.
Theorem (Pink). Let $K$ be a local field. Let $G$ be an absolutely simple connected adjoint algebraic group, whose adjoint representation $Ad_G$ is irreducible. Let $\Gamma$ be a compact subgroup of $G(K)$. We assume that (i) $\Gamma$ is Zariski-dense in $G$; (ii) the smallest closed subfield of $K$ containing the traces of the matrices $Ad_G(g)$, $g \in \Gamma$ is $K$. Then $\Gamma$ is open in $G(K)$.
This is essentially theorem 0.7 of Pink's paper, with additional hypothesis (ii) to guarantee that the subfield $E$ of $K$ in this statement is $K$ itself. The case $K=\mathbb R$ or $\mathbb C$ is essentially trivial using Lie theory, and Pink attributes it to Weyl (but note that even in this case hypotheses (i) and (ii) are necessary). If $K$ is non-archimedean, it is the field of fractions of a ring $A$ which is either $\mathbb F_q[[T]]$ or a finite extension of $\mathbb Z_p$; Those $A$ are the compact regular local domains of dimension $1$. Note that the basic examples of compact subgroups $\Gamma$ of $G(K)$ are the closed subgroups of $G(A)$ for some model of $G$ over $A$ (and all examples are of this type, see first comment below), and for these subgroups the conclusion of the theorem amounts to saying that $\Gamma$ is open in $G(A)$.
Has Pink's theorem been generalized compact local domains $A$ of higher dimensions (regular if that helps)? For example to the cases $A=\mathbb F_p[[x,y]]$, or $A=\mathbb Z_p[[x]]$? In other words, is there a criterion for a closed subgroup $\Gamma$ of $G(A)$ to be open in $G(A)$?
A statement in higher dimension will almost certainly need more hypotheses on $\Gamma$ than just (i) and (ii). I think that (i) should be reinforced by requiring that the image of $\Gamma$ is Zariski-dense in $G(Frac(A/P))$, where $P$ runs amongst non-zero prime ideals of $A$ or something like this. My question is: has this problem been studied somewhere in the literature? A mathscinet research starting with Pink's paper gave nothing, but I may have missed important things. Since Pink's theorem is fifteen years old and has been widely used ever since, it seems likely that its generalization to larger rings has been considered.
Technical remark: when $A$ is as in the question, the fraction field $K$ of $A$ has no natural field topology extending the topology of local ring of $A$, except in the dimension $1$ case. (Otherwise $K$ would be locally compact, hence a finite extension of $\mathbb Q_p$ or $\mathbb F_p((x))$ by Weil's classification of such field, and $A$ would be a dvr). This is why in formulating the question, I was careful to work over $A$, not $K$.
