EDIT 4. This post claims that the answer is No in general. I hope it's correct. END OF EDIT 4.
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.
EDIT 2. I scanned a few pages of Weil's L'intégration dans les groupes topologiques et ses applications and of Raghunathan's Discrete subgroups of Lie groups, and highlighted some statements. The highlighted statements from Raghunathan's book imply that the answer to the main question is No for semisimple Lie groups. (Of course, there might be more elementary arguments.)
[On page 100 of Raghunathan's book (one of the scanned pages) one reads "As will be seen later ... any lattice in a connected Lie group is finitely generated". Unfortunately, I haven't been able to find where, in the sequel of the book, this is proved. If somebody could indicate the appropriated page (and even scan it), it would be great!] END OF EDIT 2.
EDIT 3. Keivan Karai's answer convinced me that there are elementary arguments showing the negativity of the answer to the main question for semisimple Lie groups. END OF EDIT 3.