I am working in the context of Mazur's deformation rings, namely let $p$ be a prime number and $R$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$ and a isomorphism $R / \mathfrak{m} \simeq k$ where $k$ is a finite field extension of $\mathbb{F}_p$, the finite field with $p$ elements.
Let $G$ be a profinite group and $\rho : G \to GL_n(R)$ a representation.
Is it possible that there is a subgroup $H \subset G$ of finite index such that the restriction $\rho_{|H}$ takes its values in a strict subring $S \subset R$ and there is an element $g \in G$ such that $\rho(g) \in R \backslash S$ ?