Quartic del Pezzo surfaces over ${\mathbb{F}}_p(t)$ without quadratic points

We construct an infinite family of quartic del Pezzo surfaces over ${\mathbb{F}}_p(t)$ with no quadratic points, for all primes $p\mathrm{\ne }2$. This answers a question of Colliot–Thélène, Creutz and Viray in the negative, which asks whether every quartic del Pezzo surface has quadratic points over $C_2$ fields. We exhibit a Brauer–Manin obstruction on the variety parametrising lines associated to the quartic del Pezzo surface.