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 of such modules and hence have finite projective dimension. So what you want can't happen: there is no such $R$.

Added for clarity: the projective dimension of a module $M$ is the largest $n$ such that $Ext^n(M,-)$ is non-zero. Since Ext commutes with direct sums a direct summand of a finite direct sum of modules of finite projective dimension has finite projective dimension.

This doesn't quite answer the question but shows that the real question is for an example where all cyclic modules have finite but unbounded projective dimension. So I will leave this here.

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 of such modules and hence have finite projective dimension. So what you want can't happen: there is no such $R$.

Added for clarity: the projective dimension of a module $M$ is the largest $n$ such that $Ext^n(M,-)$ is non-zero. Since Ext commutes with direct sums a direct summand of a finite direct sum of modules of finite projective dimension has finite projective dimension.

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 $R$.

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 of such modules and hence have finite projective dimension. So what you want can't happen: there is no such $R$.

Added for clarity: the projective dimension of a module $M$ is the largest $n$ such that $Ext^n(M,-)$ is non-zero. Since Ext commutes with direct sums a direct summand of a finite direct sum of modules of finite projective dimension has finite projective dimension.