If you take the compositum $K=TP$ of the maximal tamely ramified extension $T$ of $\mathbf{Q}_p$ with a totally ramified extension $P$ of $\mathbf{Q}_p$ of degree $p$, then $K$ is *not* algebraically closed, its residue field is $\bar{\mathbf{F}}_p$, and the value group is $\mathbf{Q}$.