Let $R$ be a ring of characteristic $p$. Let $G$ be the kernel of the natural map $\pi_1^{et} (\operatorname{Spec} R[[x]] [1/x]) \to \pi_1^{et}( \operatorname{Spec} R)$. $G$ has a natural map to $\prod_{l\neq p} \mathbb Z_l(1)$, coming from the etale coverings adjoining the $n$th roots of $x$ for $n$ prime to $p$. Is the kernel of this natural map always a pro-$p$ group?

For $R$ a field, this is standard. But I'm not sure when $R$ is not a field.