We prove that the space of symplectic embeddings of $n\geq 1$ standard balls, each of capacity at most $\frac{1}{n}$, into the standard complex projective plane $\mathbb{CP}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{CP}^2$. Our techniques also suggest that for every $n \geq 9$, there may exist infinitely many homotopy types of spaces of symplectic ball embeddings.