A non discrete, torsion-free and $\sigma-$compact, locally compact abelian group?

Question

We know that every countable, discrete torsion-free group is $\sigma-$compact. Is there a non discrete, torsion-free, $\sigma-$compact, locally compact abelian group?

Answer 1

Yes, $\mathbb{R}$, with addition.

Answer 2

Both $\mathbb{R}$ and $\mathbb{S^1}$ come to mind.

Edit: just kidding. Only $\mathbb{R}$.