Is $(XY - 1)$ a maximal ideal in $k[[X]][Y]$?

Is $(XY - 1)$ a maximal ideal in $k[[X]][Y]$, and if so, how can I see it?

It is at least prime because the generator is irreducible, and by the same argument it is maximal among all principal ideals. But I haven't gotten further than that - Finding units in the quotient ring didn't turn out well, either.

If $A$ is a commutative ring and $s\in A$ any element , the ring $ B=A[Y]/(sY-1)$ is isomorphic to the localized ring $S^{-1}A$, where $S$ is the multiplicative set $S=\lbrace 1,s,s^2,s^3,... \rbrace \subset A$.
The prime ideals of the localized ring $B$ are in bijection with the prime ideals of $A$ disjoint from $S$: this is perhaps the fundamental fact about localization.

In your case $A=k[[X]]$, $s=X$ and the prime ideals of the DVR $A=k[[X]$ are $(0)$ and $M=(X)$.
The only surviving prime ideal in $B=S^{-1}A$ is the zero ideal, since obviously $M$ is killed by localization, and thus $B$ is a field (called the field of Laurent series k((X)) over $k$).
If you go back to the other definition of $B$ as $B=k[[X]][Y]/(XY-1)$, you see that your ideal $(XY-1)$ was indeed maximal.