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.