Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?

More generally, I'm looking for a criterion for locally compact Hausdorff topological groups to not contain a nested exhaustive sequence of compact subgroups (i.e. locally compact Hausdorff topological groups $G$ with a countable family of nested compact subsets $K_n$ such that $\bigcup_{n\ge 0}K_n = G$).

**Example** Real and complex Lie groups are $\sigma$-compact exactly when they are compact.

**Example** The same group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, endowed with the discrete topology. It is uncountable, so it's not $\sigma$-compact when endowed with the discrete topology, if I'm not missing anything.

subspaces, which seems very different to an increasing sequence of compactsubgroups. I think many non-compact Lie groups are $\sigma$-compact. $\endgroup$