As discussed in the question number fields with no unramified extensions, it is an open question whether there are an infinite number of number fields that have no unramified extensions. Inspired by this, I would like to ask the weaker question as follows:

**Question:**

- Let $p$ be a fixed prime. Are there infinitely many number fields such that its maximal unramified $p$-extension is finite?
- Is there a prime $p$ such that $\mathbb{Q}(\zeta_{p^m})$ admits a finite maximal unramified $p$-extension for all positive integer $m$?

**Update:** the answer for question $1$ is Yes. In Theorem I of Construction of maximal unramified p-extensions with prescribed Galois groups by Manabu Ozaki, it's proved that

for any prime $p$ and any given finite $p$-group $G$, there exists a number field $F$ such that the Galois group of the maximal unramified $p$-extension over $F$ is isomorphic to $G$.

Since there are infinitely many finite $p$-groups, we see that there are infinitely many number fields such that its maximal unramified $p$-extension is finite.