The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-group for any prime $q$.
In particular, the smallest example of this form is $\operatorname{BS}(3, -3) = \langle a, b \mid ba^3b^{-1}a^3 = 1 \rangle$.
(Note that OP's question was posed by McCarron in 1996).
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[1] Moldavanskii, D. I., Residual nilpotence of groups with one defining relation, +; translation from Mat. Zametki 107, No. 5, 752-759 (2020). ZBL07215590.