  The behavior of the Frobenius map is investigated for valuation rings of
prime characteristic. We show that valuation rings are always F-pure. We
introduce a generalization of the notion of strong F-regularity, which we call
F-pure regularity, and show that a valuation ring is F-pure regular if and only
if it is Noetherian. For valuations on function fields, we show that the
Frobenius map is finite if and only if the valuation is divisorial; in this
case the valuation ring is Frobenius split. For Noetherian valuation rings in
function fields, we show that the valuation ring is Frobenius split if and only
if Frobenius is finite, or equivalently, if and only if the valuation ring is
excellent.
