This is an open problem; I personally suspect it cannot happen.
More generally let $X$ be a smooth projective variety over a field $k$ which is finitely generated over its prime subfield. Then Skorobogatov and Zarhin ask in Question 2 of [1] whether $\mathrm{Br}(\bar{X})^\Gamma$ is in fact always finite.
It is known that the Tate conjecture implies finiteness of the $\ell$-primary part of this group for all primes $\ell$. You can read more about this in Section 16.1 of [2]. The real challenge is to therefore show that the $\ell$-primary part is actually trivial for all but finitely many primes $\ell$; this is closely related to the Tate conjecture but does formally follow from it.
[1] Skorobogatov, Alexei N.; Zarhin, Yuri G. A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces. J. Algebraic Geom. 17 (2008), no. 3, 481--502.
[2] Colliot-Thélène, Jean-Louis; Skorobogatov, Alexei N. The Brauer-Grothendieck group. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 71. Springer, Cham, [2021].