Non-uniqueness of certain Hahn–Banach extensions

Let $f$ be a continuous linear functional defined on a subspace $M$ of a normed space $X$. If $X$ is real or complex, there are results that characterize uniqueness of continuous extensions $F$ of $f$ to $X$ for every subspace $M$ and those that apply just to $M$. If $X$ is defined over a non-Archimedean valued field $K$ and the norm also satisfies the strong triangle inequality, the Hahn–Banach theorem holds for all subspaces $M$ of $X$ if and only if $K$ is spherically complete and it is well-known that Hahn–Banach extensions are never unique in this context. We give a different proof of non-uniqueness here that is interesting for its own sake and may point a direction in which further investigation would be fruitful.