In Lang's book "Algebra", theorem 9.2, it said that suppose $f\in \mathbb{C}[[X,Y]]$, then by some conditions imposed to $f$, $f$ can be written as a product of a polynomial $g\in \mathbb{C}[[X]][Y]$ and a unit $u$ in $\mathbb{C}[[X,Y]]$.
It suggests the following claim is not true in general.
Let $f\in \mathbb{C}[[X,Y]]$, then there exists a $g\in \mathbb{C}[X,Y]$ and a unit $u\in \mathbb{C}[[X,Y]]$ such that $f=gu$.
I would like to find a counter-example.
Thanks.