  For any $g_1, g_2 \ge 0$, this paper shows that there is a cocompact lattice
$\Gamma < \mathrm{PU}(2,1)$ such that the ball quotient $\Gamma \backslash
\mathbb{B}^2$ is birational to a product $C_1 \times C_2$ of smooth projective
curves $C_j$ of genus $g_j$. The only prior examples were $\mathbb{P}^1 \times
\mathbb{P}^1$, due to Deligne--Mostow and rediscovered by many others, and a
lesser-known product of elliptic curves whose existence follows from work of
Hirzebruch. Combined with related new examples, this answers the rational
variant of a question of Gromov in the positive for surfaces of Kodaira
dimension $\kappa \le 0$, namely that they admit deformations $V^\prime$ such
that there is a compact ball quotient $\Gamma \backslash \mathbb{B}^2$ with a
rational map $\Gamma \backslash \mathbb{B}^2 \dashrightarrow V^\prime$. Often
the proof gives the stronger conclusion that $V^\prime$ is birational to a ball
quotient orbifold. All examples are shown to be arithmetic, and even
arithmeticity of Hirzebruch's example appears to be new.
