This question is not directly related to, but was inspired by, [this question](https://mathoverflow.net/questions/77583/is-the-free-product-of-arbitrarily-many-copies-of-mathbbz-and-mathbb).  We know that a finitely generated residually nilpotent group is residually of prime-power order.  However, we may need to use different primes for different elements.  Classes of groups for which residual nilpotence forces there to be a single prime that will do for all elements (i.e., for which the group in question must be residually $p$-finite, for some $p$) seem to be interesting, and include, for instance, free products of cyclic groups.

<b>Is there a (non-cyclic) one-relator group that is residually nilpotent, but is not residually a finite $p$-group, for any prime number $p$?</b>

Such a group must be torsion-free, with trivial centre.