This article investigates an explicit description of the Baum–Connes assembly map of the wreath product $\Gamma =F\wr {\mathbb{F}}_n={\oplus }_{{\mathbb{F}}_n}F⋊{\mathbb{F}}_n$, where $F$ is a finite and ${\mathbb{F}}_n$ is the free group on $n$ generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space $\underline{\mathrm{E}}\Gamma$. As a result of our computations, we obtain that ${\mathrm{K}}_0\left({\mathrm{C}}_{\mathrm{r}}^{\ast }\right(\Gamma \left)\right)$ is the free abelian group of countable rank with a basis consisting of projections in ${\mathrm{C}}_{\mathrm{r}}^{\ast }\left({\oplus }_{{\mathbb{F}}_n}F\right)$, and ${\mathrm{K}}_1\left({\mathrm{C}}_{\mathrm{r}}^{\ast }\right(\Gamma \left)\right)$ is the free abelian group of rank $n$ with a basis represented by the unitaries coming from the free group.