Is there a Dedekind domain $B$ satisfying the following two conditions:
- $B$ is an algebra over a finite quotient PID $A$ such that $B$ is free of finite rank as a module over $A$;
- $B$ has infinite class group?
Here a finite quotient PID is a principal ideal domain all of whose proper quotients are finite rings.
It is known that if we demand in addition that $A$ has "enough elements of small norm" and its ideal norm satisfies a weakened form of the triangle inequality, then no such $B$ exists (see this MO answer).
Papers by Goldman and Heitmann mentioned in the answers to this MO question, construct Dedekind domains with finite quotients and with infinite class group. These are overrings of $\mathbb{Z}[X]$, so are not finite over $\mathbb{Z}$, but I do not know whether they are finite over any other PID.
On the other hand, Leedham-Green showed that for every abelian group $G$ there is a PID $A$ with fraction field $K$ and a separable quadratic extension $L/K$ such that the integral closure of $A$ in $L$ has class group isomorphic to $G$. He thus constructs Dedekind domains which are finite over a PID, but have infinite class group. However, as far as I can tell, he is able to do this because the ground field is "very large", like an algebraic closure. In particular, the PID is far from having finite quotients. The same seems to apply to Pete L. Clark's proof of Leedham-Green's version of Claborn's theorem.
From these papers I got the impression that it is difficult to find a Dedekind domain with infinite class group which has both finite quotients and is finite over a PID. But given that there is no known reason why such a Dedekind domain should not exist, it is perfectly possible that some variation of these constructions might produce such as example.