This relates to this question: Existence of maximal totally ramified extensions of an arbitrary CDVF

Let $K$ be a local field with finite residue field of characteristic $p>0$. Does there exist a maximal totally ramified $p$-extension of $K$? In other words, if $K^{tame}$ is the maximal tamely ramified extension of $K$, does the short exact sequence $$ 1 \longrightarrow Gal(K^{sep}/K^{tame}) \longrightarrow Gal(K^{sep}/K) \longrightarrow Gal(K^{tame}/K) \longrightarrow 1 $$ split? My guess would be that it doesn't, but maybe there is some trick I am missing here.