A finiteness theorem for the Brauer group of K3 surfaces in odd   characteristic
  Let $k$ be a field finitely generated over the finite field $\mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)\to Br(X)$ is finite. 
Alexei N. Skorobogatov and Yuri G. Zarhin
math.AG math.NT