$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$.

My questions are: Is it possible to have an example of a commutative ring with unity $R$ such that $R \cong R[X,Y]$ but $R \ncong R[X]$, where

(i) $R$ is UFD ?

(ii) $R$ is a Valuation ring ?

UPDATE : Since $R[X]$ always has infinitely many maximal ideals for any commutative ring with unity $R$, so $R \cong R[X,Y]$ implies $R$ can't be local, hence question (ii) is meaningless. Question (i) still remains ...