  We prove that the $p^\infty$-torsion of the transcendental Brauer group of an
abelian variety over a finitely generated field of characteristic $p>0$ is
bounded. This answers a (variant of a) question asked by Skorobogatov and
Zarhin for abelian varieties. To do this, we prove a "flat Tate conjecture" for
divisors. In the text, we also study other geometric Galois-invariant
$p^\infty$-torsion classes of the Brauer group which are not in the
transcendental Brauer group. These classes, in contrast with our main theorem,
can be infinitely $p$-divisible. We explain how the existence of these
$p$-divisible towers is naturally related to the failure of surjectivity of
specialisation morphisms of N\'eron--Severi groups in characteristic $p$.
