Is there any example of a topologically finitely generated profinite group $G$ which is nilpotent as a group but has an infinite torsion-part?
The question is motivated by the fact that an algebraic finitely generated nilpotent group has a finite torsion-part and I known that such is also the case for any topologically finitely generated profinite group which is abelian. I'm just curious since it seems I cannot find anything online in one sense or in another.