Fix a prime $ p $. We call an infinite profinite group $ G $ a Fontaine-Mazur group (with respect to $ p $) if every continuous homomorphism $ G\to {\rm GL}_{n}(\overline{\mathbb{Q}_{p}}) $$ G\to {\rm GL}_n(\overline{\mathbb{Q}}_p) $ has finite image for any positive integer $ n $. The motivation for defining such a group comes from the following example:
Let $ K $ be a number field, $ S $ a finite set of primes of $ K $ not containing any primes above $ p $ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ unramified outside $ S $. Then the unramified Fontaine-Mazur Conjecture claims that $ G_{K,S} $ is a Fontaine-Mazur group (with respect to $ p $), cf. Conjecture 5a in [Fontaine and Mazur. "Geometric Galois representations." Elliptic curves, modular forms, Fermat’s last theorem (Hong Kong, 1993).]
My question is that: can someone give more examples of Fontaine-Mazur groups? For example, is $ {\rm SL}_{n}(\mathbb{F}_{p}[[T]]) $$ {\rm SL}_n(\mathbb{F}_p[[T]]) $ a Fontaine-Mazur group? More generally, for any infinite complete Noetherian local $ \mathbb{F}_{p} $$ \mathbb{F}_p $-algebra $ A $, is $ {\rm SL}_{n}(A) $$ {\rm SL}_n(A) $ a Fontaine-Mazur group?
