Let $k$ be a field with a non-archimedean absolute value. If $k$ is complete, then two norms on a finite dimensional $k$-vector space are always equivalent. This fact is for example commonly invoked in the proof that a complete non-archimedean field is henselian.
The field $k$ is called henselian if for every algebraic field extension $K/k$, the absolute value of $k$ extends uniquely to $K$.
If we only require $k$ to be henselian, is it still true, that any two norms on a finite dimensional vector space over $k$ are equivalent?
It is well known that if we don't impose any restriction on the absolute value, then the conclusion is wrong, but I am not aware of any counter examples in the henselian case.