Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?

Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the cohomological Brauer group of a quasi-compact scheme is torsion) exploiting the Kummer sequence. For integral affine schemes and the multiplicative group one can also prove this using that the Brauer group injects into the Brauer group of the fraction field, which is torsion as a Galois cohomology group.

Let $A/X$ be an abelian scheme. Is $H^n(X,A)$ torsion for $n > 0$?

Perhaps this can be proved analogously as Proposition IV.2.7 of Milne's Étale cohomology (where it is proved that the cohomological Brauer group of a quasi-compact scheme is torsion) exploiting the Kummer sequence.