Do Gorenstein rings necessarily have finite projective dimensions?



Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)?
Take $A=k[x]/(x^2)$ for a field $k$. This is a self-injective $k$-algebra (that is, it is an injective module over itself), so it is Gorenstein. Yet the residue field is of infinite projective dimension.


