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 [2] in 1996).

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[1] <cite authors="Moldavanskii, D. I.">_Moldavanskii, D. I._, [**Residual nilpotence of groups with one defining relation**](https://doi.org/10.1134/S0001434620050090), translation from Mat. Zametki 107, No. 5, 752-759 (2020). [ZBL07215590](https://zbmath.org/?q=an:07215590).</cite>

[2] <cite authors="McCarron, James">_McCarron, James_, [**Residually nilpotent one-relator groups with non-trivial centre**](https://www.ams.org/journals/proc/1996-124-01/S0002-9939-96-03148-6/S0002-9939-96-03148-6.pdf), Proc. Amer. Math. Soc. 124, No. 1 (1996). </cite>