Is there any ring $R$ of infinite global dimension such that any $R$-module is a retract (i.e. direct summand) of some $\oplus_{i\in I}M_i$ where each $M_i$ has finite projective dimension?

I ask this because in the easy examples of rings of infinite global dimension I have in mind, there is always a simple $R$-module with infinite projective dimension. I wanted to know if this happens in general, or if pathological examples exist. In the situation I ask for above, the global dimension would be infinite, morally for asymptotic reasons.