In https://projecteuclid.org/download/pdf_1/euclid.nmj/1118799684 Auslander shows that the global dimension of $R$ is the supremum of the projective dimensions of cyclic modules.
If a cyclic module is a retract of a direct sum of modules of finite projective dimension, then it would be a retract of a finite direct sum such modules and hence have finite projective dimension. So what you want can't happen: there is no such $R$.