# Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of $S_p$, the $t$-adic completion of $F_p\{t\}$, the field of formal Laurent series over the field of $p$ elements. Then is $F$ injective with respect to topological embeddings in the category of normed linear $F$-spaces?

There exists a $p$-adic Hahn Banach theorem.
Let $K$ be a spherically complete ultrametric field. $E$ a $K$-normed space and $V$ a subspace of $E$. For every bounded linear functional $\phi$ defined on $V$, there exists a bounded linear functional $\psi$ defined on $E$ whose restriction to $V$ is $\phi$ and such that $\|\phi\|=\|\psi\|$.