Example of Artinian local ring over a field that is not finitely-dimensional

Question

Let $A$ be an Artinian local ring over a field $k$. All the examples I know are finite-dimensional over $k$, like $k[x]/(x^n)$. Is it true in general?

Update. Well, $A$ probably should be finitely-generated, because it is a quotient of a local ring of a variety.

Answer 1

The answer to the modified question is affirmative: The residue field $K:=A/{\mathfrak m}$ of $A$ is a finitely generated, hence finite field extension of $k$, and the finitely many non-zero filtration quotients ${\mathfrak m}^l/{\mathfrak m}^{l+1}$ are of finite dimension over $K$, hence also over $k$.

Answer 2

No: $\mathbb Q\subset \mathbb C$
(This answers the original question, which didn't mention finite generation. Hanno has perfectly answered the modified question)