A Noetherian local ring $R$ admits a small Cohen-Macaulay (CM) algebra if there is an injective map of rings $R\hookrightarrow S$ such that every system of parameters of $R$ becomes a regular sequence in $S$ and $S$ is a finite $R$-module.

If $R$ is a non CM normal domain containing the rationals, then $R$ cannot admit a small CM algebra. This is because there exists a retraction from $S\rightarrow R$ using the trace map corresponding to the fraction fields.

My question: Is an example of the failure of this non existence known in mixed characteristic $p>0$? More precisely, is there a concrete example of a Noetherian local non CM normal domain $R$ of mixed characteristic $p>0$ such that $R$ admits a small CM algebra? Thank you.

Please note: I am a graduate student and would like to know if such examples are already known. I believe I am able to construct examples but I am not sure whether this is new.