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Here is another example with easier groups. Formanek showed that the group ring over any field of a free product of non-trivial groups is primitive, has a faithful simple module. That means that $\mathbb F_pG$ is primitive whenever $G=A\ast B$. And of course $G$ is residually finite if both $A$ and $B$ are. Obviously a faithful simple $\mathbb F_pG$-module is cyclic as a $\mathbb ZG$-module and since $G$ is infinite, it cannot be finite. So this gives lots of examples including the free group on two generators.