The answer is in the negative:

every finitely generated, residually finite, infinite torsion group G in which every element has order a power of a fixed prime p is a counter example.

Indeed, such a group is not soluble as every finitely generated soluble torsion group is finite and it cannot contain a  non-abelian free subgroup. Moreover, the group, being residually finite and a p-torsion group, is residually a finite p-group, hence residually nilpotent and thus abelian by residually nilpotent.

So it suffices to find a finitely generated, residually finite, infinite torsion group G in which every element has order a power of a fixed prime p. Such a group has been exhibited  by N. Gupta and S. Sitki in Math. Z. 182 (1983), 385–388.

Thanks Ralph for your quick answer. Do you know a counterexample if I also impose the conditions of torsion-free and finitely generated?