Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ \mathfrak{p}$ is an isomorphism.
Is $A$ necessarily a finitely generated $k$-algebra?
My first thought was that the direct product $\prod_{i \in \mathbb{N}} k$ should be an obvious counter example. But I think that any non-principal ultra filter on $\mathbb{N}$ gives us a prime ideal $\mathfrak{p}$ such that the residue field is not $k$.
Edit: So since there are counter examples for finite fields (see comments), I want to assume that the characteristic of $k$ is zero.