Recently in some research I came to the point where the strength of my conclusion bottlenecks at my ability to precisely address this question:
Let $R$ be a ring such that $R[[x]]$, the ring of formal power series with coefficients in $R$, is a GCD domain (whatever that entails for $R$). What additional properties (if any) do $R$, $R[x]$, or $R[[x]]$ need to possess so that elements relatively prime in $R[x]$ are also still relatively prime in $R[[x]]$?
With a little effort we can show (by way of a weak Bezout-type identity that holds for any polynomial GCD domain) that it's sufficient for $R[[x]]$ to be atomic in addition to a $GCD$ domain (hence a $UFD$) — or equivalently it is sufficient for $R$ to satisfy the $ACCP$. This is an OK result, but I'm hoping that there's some slack here. In particular, since $R[[x]]$ a $GCD$ already implies that $R$ is Archimedean, I wonder if there's any point in the ground between Archimedean and $ACCP$ where $R$ is still structured enough for coprimeness to lift from $R[x]$ to $R[[x]]$.
And if there isn't, I'd love to better understand where the limitation is coming from!
Update. If the answer below checks out, then it's indeed enough for $R$ to be Archimedean (along with being a GCD domain).
Actually if $R[[x]]$ is a PID, you can use the Bézout lemma. If $f,g\in R[[x]]$ are coprime than exist $h,k\in R[[x]]$ such that $h·f+k·g=1$ this equality holds also in $R[x]$, in this direction you don't even need again Bézout lemma (a common divisor of $f$ and $g$ in $R[x]$ would also divide $1$). I don't see any mistake in this reasoning, is there?