Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite?
In other words, let $f$ be a non-zero element of $\mathbb Z_ \ell$. Is the quotient module $\mathbb Z_ \ell/ (f)$ finite?
My guess is that we can assume $f= \ell^n$ for some non-negative $n$ (as units don't change the quotient). Then the quotient module $\mathbb Z_ \ell/ (f)$ should just be $\mathbb Z/(\ell^n)$. Is that correct? What triviality am I missing?
Your reasoning seems correct to me. To add more details, recall the following fact:
Proposition: If $R$ is Noetherian, $I$ an $R$-ideal, $M$ a f.g. $R$-module, then $\widehat{M}/I^i\widehat{M} \cong M/I^iM$, where $\widehat{}$ denotes $I$-adic completion.
Now $\mathbb{Z}_l$ is the $(l)$-adic completion of $\mathbb{Z}$ or $\mathbb{Z}_{(l)}$ (the localization at $(l)$), hence is a DVR with uniformizer $l$, so as you say any nonzero $f \in \mathbb{Z}_l$ is a unit times $l^n$, and $f \mathbb{Z}_l = l^n \mathbb{Z}_l$. (To make the reduction to $\mathbb{Z}_l/f\mathbb{Z}_l$, we use the structure theorem for f.g. modules over a PID).