Let $\Gamma$ be a discrete group.
It is well known that if $\Gamma$ is amenable, then its full $C^*$ algebra $C^*(\Gamma)$ is exact. Moreover, if $\Gamma$ is residually finite then the converse also holds: if $C^*(\Gamma)$ is exact then $\Gamma$ is amenable (e.g. proposition 3.7.11. Brown-Ozawa "$C^*$-algebras and finite dimensional approximations").
Do we need to asumme that $\Gamma$ is residually finite? Or in other words, is there a discrete group $\Gamma$ which is not amenable but has exact full $C^*$ algebra $C^*(\Gamma)$?