  We construct a torsion-free arithmetic lattice in
$\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$
arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental
group of a square complex with universal covering $T_3\times T_3$, a product of
trees with constant valency $3$, which has minimal Euler characteristic.
Furthermore, our lattice gives rise to a fake quadric over
$\mathbb{F}_2(\!(t)\!)$ by means of non-archimedean uniformization.
