I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group.

Related questions: Schemes with trivial brauer group and Brauer group of projective space

The affine plane $\mathbf{A}^2_k$ should be one example. Edit: I am especially interested in cases where the surface is *not rational*.

Note that it is difficult to pass from *projective* surfaces to affine surfaces because of purity for Brauer groups: $0 \to \mathrm{Br}(S)(\ell) \to \mathrm{Br}(S - C)(\ell) \to H^1(C,\mathbf{Q}_\ell/\mathbf{Z}_\ell)$. Here, $H^1(C,\mathbf{Q}_\ell/\mathbf{Z}_\ell) = 0$ iff $C$ has genus $g(C) = 0$, is affine and $\bar{C} - C$ (the divisor at infinity) is a one-point set.