Continuity of K-theory: An example in equal characteristics

If k is a perfect field of characteristic p > 0, we show that the Quillen K-groups K-i(k[[t]]) are uniquely p-divisible for i = 2,3. In fact, the Milnor K-groups K-n(M)(k((t))) are uniquely p-divisible for all n > 1. This implies that K(A) --> holim((n) over left arrow) K(A/m(n)) is 4-connected after profinite completion for ii a complete discrete valuation ring with perfect residue field.