Is there a finite quotient Dedekind domain with infinitely many primes of small norm?

• If $char(A)=0$ : $N(I) \le d$ implies $A/I$ is a quotient of $A/(d!)$, your assumption is that it is a finite ring with finitely many quotients thus only finitely many ideals with $N(I) \le d$

• If $char(A) = p$ : look at the ideal $J_m = \sum_{a \in A} (a^{p^m}-a) A$,

  assume this time $I$ is a prime ideal,

  $N(I) \le p^k$ implies $A/I$ is a quotient of $A/J_{k!}$, again your assumption is that it is a finite ring with finitely many quotients thus only finitely many ideals with $N(I) \le p^k$

That is to say the next step is to ask what happens if $A/(d!)$ or $A/J_{k!}$ is not a finite ring