Milnor $K$-theory of $p$-adic rings

We study the mod $p^r$ Milnor $K$-groups of $p$-adically complete and $p$-henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod $p^r$ Gersten conjecture for Milnor $K$-theory locally in the Nisnevich topology. In characteristic $p$ we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.