Is there a non-artinian noetherian ring whose non-units are zero-divisors?

Equivalent formulation:

Is there a noetherian ring of positive dimension whose non-units are zero-divisors?

[In this post, "ring" means "commutative ring with one", and "dimension" means "Krull dimension".]

Here is the motivation:

Let $A$ be a ring whose non-units are zero-divisors.

If $A$ is *not* noetherian, then $A$ can have positive dimension: see this answer of user18119.

If $A$ is noetherian and *reduced*, then $\dim A\le0$: see this answer of user26857.

[Recall that a noetherian ring is artinian if and only if its dimension is $\le0$. Recall also that a ring has the property that its non-units are zero-divisors if and only if it is isomorphic to its total ring of fractions.]