On D. I. Moldavanskii's question about $p$-separable subgroups of a free group

We prove that every free nonabelian group has a finitely generated isolated subgroup not separable in the class of nilpotent groups. This enables us to give a negative answer to the following question by D. I. Moldavanskii in the “Kourovka Notebook”: Is it true that every finitely generated $p'$-isolated subgroup of a free group is separable in the class of finite $p$-groups?