Here is another example with different groups.  [Formanek][1] showed that the group ring over any field of a free product of non-trivial groups (and not both order 2) 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.


  [1]: https://www.sciencedirect.com/science/article/pii/0021869373900112