  Let $\mathbb{F}_q$ be the finite field with $q$ elements,
$F:=\mathbb{F}_q(T)$ and $F^{\operatorname{sep}}$ a separable closure of $F$.
Set $A$ to denote the polynomial ring $\mathbb{F}_q[T]$. Let $\mathfrak{p}$ be
a non-zero prime ideal of $A$, and $\mathscr{O}$ be the completion of $A$ at
$\mathfrak{p}$. Given any integer $r\geq 2$, I construct a Galois
representation $\rho:\operatorname{Gal}(F^{\operatorname{sep}}/F)\rightarrow
\operatorname{GL}_r(\mathscr{O})$ which is unramified at all non-zero primes
$\mathfrak{l}\neq \mathfrak{p}$ of $A$, and whose image is a finite index
subgroup of $\operatorname{GL}_r(\mathscr{O})$. Moreover, if the degree of
$\mathfrak{p}$ is $1$, then $\rho$ is also unramified at $\infty$.
