One counterexample is $K\rtimes K^*$ when $K$ is an infinite field of characteristic $p>0$. This is not residually finite (because $K$ is mapped injectively or trivially in every quotient), while $K$ is residually finite.
The group $K^*$ may fail to be residually finite (e.g., when $K$ is algebraically closed), but for $K=\mathbf{F}_p(t)$ it is residually finite (direct product of $\mathbf{F}_p^*$ with a free abelian group of infinite rank).