Is there a finite dimensional local ring with infinitely many minimal prime ideals?

Equivalent formulation:

Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of minimal prime sub-ideals of $\mathfrak p$ is infinite?

Here "ring" means "commutative ring with one", "dimension" means "Krull dimension", and "local ring" means "ring with exactly one maximal ideal" (warning: some authors call "quasi-local ring" a ring with exactly one maximal ideal, and "local ring" a **noetherian** ring with exactly one maximal ideal; it is well known that a noetherian ring has only finitely many minimal prime ideals).