Splitting of $k[[x_1,...,x_n]]\to \bar k [[x_1,...,x_n]]$

Question

Let $k$ be a field, of positive characteristic, with algebraic closure $\bar k$. Then, is it true that for every integer $n\ge 1$, the inclusion map $k[[x_1,...,x_n]]\to \bar k [[x_1,...,x_n]]$ splits as a map of $k[[x_1,...,x_n]]$-modules?