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 $K^{M}_{n}(k((t)))$ are uniquely $p$-divisible for all $n>1$. This implies that $K(A)\to {\operatornamewithlimits{holim}}_{\overleftarrow {n}} K(A/\mathfrak{m}^{n})$ is $4$-connected after profinite completion for $A$ a complete discrete valuation ring with perfect residue field.