Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $\bar k$ that is unramified outside $S$, and put $$G = \mathrm{Gal}(k_S / k).$$
We let $C_S$ be the direct limit, over all finite extensions $K$ of $k$ inside $k_S$, of the $S$-idele class group $C_S(K)$. For a prime number $p$, we let $C_S[p]$ be the (topological $G$-module) of all elements in $C_S$ whose order divides $p$.
Is the Pontryagin dual of $C_S[p]$ a topologically finitely generated $G$-module?
The Pontryagin dual here is just $\mathrm{Hom}(C_S[p], \mathbb{F}_p)$.
Notation and definitions are taken from `Cohomology of Number Fields' by Neukirch, Schmidt, and Wingberg.
