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?