Dichotomy for the p-primary Brauer-Manin obstruction in characteristic p

Let X be a smooth, projective, geometrically integral variety over a global function field k of characteristic p. We show that if the unipotent Brauer group of X is zero, and the Picard scheme of X is p-torsion free, then only finitely many places of k can be potentially relevant to the p-primary Brauer-Manin obstruction for X. On the other hand, if the unipotent Brauer group is non-zero, then almost all places of k are potentially relevant. If X is base changed from a finite field, then in both cases the finite set of exceptional places is empty.