The posting of this question was suggested by Yemon Choi: see Discrete cyclic subgroup.. The question is not mine; it's just a rephrasing of Discrete cyclic subgroup.

By page 110 of Weil's book L'intégration dans les groupes topologiques et ses applications, the answer is No in the abelian case.

I know almost nothing about locally compact groups. The question might be very easy for experts, and perhaps even for laymen. In the unlikely event the question is difficult, here is a particular case:

Let G be a non-compact connected Lie group. Does G admit a discrete infinite cyclic subgroup?

EDIT 1. I think that, by known results about lattices, the answer is No for semisimple Lie groups. Thanks for correcting me if I'm wrong, or (even better) for providing precise statements and references. Again, plenty of MathOverflowers know this stuff much better than I. I'm making it a Community Wiki. END OF EDIT 1.