Let $A$ be a Poisson $\mathbb{C}$-algebra: $A$ has a the structure of a complex commutative algebra and at the same time it carries the structure of a Lie algebra, with Lie bracket $\{\cdot, \cdot\}$. The latter should be a biderivation of the commutative multiplication, as well as the Lie multiplication.

Just as the definition of an $A$-module may be seen as an abstraction of left multiplication of $A$ on itself, we can define a Poisson $A$-module to be a vector space $V$ together with two multiplicative rules $A \times V\rightarrow V$ satisfying compatibility conditions. For more detail see one of the original papers by Farkas.

My question is the following:

**Is every simple Poisson module finitely generated as an $A$-module?**

I expect that the answer is `no' although I have been unable to find a counterexample or indeed any reference which considers this question. I am especially interested in the case where $A$ is an affine algebra, ie. finitely generated and reduced. Proposition 1.1 in the paper cited above states that when $A$ is symplectic every Poisson module is induced by a $\mathcal{D}$-module, so this may be a fertile source of counterexamples: is every simple $\mathcal{D}(A)$-module finitely generated over $A$?

Thanks in advance!

**Edit**: Following Ben's response I am now specifically looking for examples where $A$ is a reduced Poisson algebras, ie. no nilpotent elements.