Algebra switching – Example of a local ring that is not CM and an MCM module on it

I am looking for an example of a local commutative noeterian ring $ (A, m) $,
and a maximum module of Cohen-Macaulay $ M $ more than $ A $ (in particular $ M $ is finely generated on $ A $), as for some $ p in Spec (A) $we have that $ M_p = 0 $,
and $ A_p $ Nor is it a Cohen-Macaulay ring. Are there any such examples?