It is quite difficult to google for non-abelian torsion-free groups.
I am primarily interested in metrizable (''not too big'') compact groups. However, even if we drop the metrizability condition I do not know the answer.
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1The Heisenberg group over the $p$-adic ring $\mathbf{Z}_p$. – YCor Mar 22 '22 at 22:56
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@YCor Thanks! That is correct. I posted a question on MO that tells more about my motivation. – chj Mar 28 '22 at 16:00