There may be a section in the old Curtis and Reiner "Theorems of Blichfeld, Burnside and Frobenius" which answers this to a large extent, although they don't think about $p$-adics.
Thanks to Jordan's theorem, there is an Abelian normal subgroup of (fixed) bounded index. So the question reduces to limiting the size of Abelian (normal) subgroups. But for the rings $R$ you discuss, there are only finitely many roots of unity in any such $R,$ and for any given $R,$ there is an explicit bound on the number of roots of unity in $R$, and an explicit bound on the size of Abelian subgroups (in fact, for a primitive absolutely irreducible group, the largest normal Abelian subgroup consists of scalar matrices).

An alternative approach is to note that the kernel of any reduction (mod $p$) (strictly, reduction (mod the appropriate prime ideal containing $p$) of a finite subgroup is a finite normal $p$-group, and the image group is a subgroup of a finite ${\rm GL}(n,q),$ ($q$ a fixed -in terms of $R$-power of $p$). There are many ways to obtain explicit bounds on the size of the normal $p$-subgroup- over some extension field, it is a monomial group, etc.

In view of questions below, let me expand a little. It is commonplace in modular representation theory to work with a $p$-modular system, which is a triple $(\mathbb{K},R,F)$ such that $R$ is a complete discrete valuation ring of characteristic $0$ such that $R$ has field of fractions $\mathbb{K},$ and $F$ is the residue field $R/J(R)$. This triple is usually taken to be large enough for $\mathbb{K}$ to contain a splitting field for a finite group $G$ and its subgroups ( for example, by assuming that $R$ contains a primitive $|G|$-th roots of unity, which we now do). It is also commonplace to identify $\mathbb{C}$-valued characters of $G$ with characters afforded by finite dimensonal $\mathbb{K}G$-modules. Details are often glossed over, but this is all perfectly permissible.

Any character of a finite dimensional $\mathbb{C}G$-module is afforded by some $\mathbb{Q}[\omega]G$-module, where $\omega$ is a primitive $|G|$-th complex root of unity. This module is determined uniquely up to isomorphism by its character. Since $\mathbb{Q}[\omega]$ is isomorphic to a subfield of $\mathbb{K}$ under current assumptions, every $\mathbb{C}G$-module ``comes from" a $\mathbb{K}G$-module. Conversely, every character afforded by a $\mathbb{K}G$-module may be decomposed uniquely using the standard inner product on the character ring into a sum of complex irreducible characters, each of which may be afforded by a $\mathbb{Q}[\omega]G$-module. So for most purposes, there is little difference between studying $\mathbb{C}G$-modules and $\mathbb{K}G$-modules, and, in particular, the maximum index of an Abelian normal subgroup of a finite subgroup of ${\rm GL}(n,\mathbb{K})$ can be no larger than the corresponding bound for ${\rm GL}(n,\mathbb{C}).$ As mentioned in comments, if one works with primitive irreducible groups, all Abelian normal subgroups are central. But large groups of scalar matrices can't be finessed, though Jordan's theorem in the primitive case shows that this is the only real obstacle to an absolute bound.