Let $R$ be a discrete valuation ring with field of fraction $K$ and residue field $k$ and let $K'$ be a finite and separable extension of $K$. If $R$ is henselian ("Hensel's lemma holds", e.g. if it is complete) and $k$ is algebraically closed then the extension $K'|K$ is totally ramified (from Hensel's lemma if follows that there's just one prime lying above the prime of $R$ and since $k$ is algebraically closed there's no residual extension).
What happens if we assume only that $R$ is strictly henselian (i.e. that $k$ is only separably closed)?
Equivalently: does it exist a finite separable extension $K'$ of $K$ such that the residual extension is a nontrivial inseparable extension? I believe that such an extension should exist. Moreover, can we find a Galois extension with this property? Can we do this both in characteristics $(0,p)$ and $(p,p)$?