A well-known theorem of Gruenberg implies that a finitely generated residually torsion-free nilpotent group is residually $p$-finite for all primes $p$. What about the converse?

**Question:**
Are there any examples of a finitely generated group $G$ with the properties that $G$ is residually $p$-finite for all primes $p$, but $G$ is not residually torsion-free nilpotent?