Let $(K,v)$ be a field equipped with a non-archimedean valuation $v,$ and let $(L,w)$ be a finite extension of $(K,v)$ with $w$ of course non-archimedean. If we denote by $K_v,L_w$ the completions of $K$ resp. $L$ at $v$ and $w$ respectively, it is easy to see that $L$ is dense in $L_w,$ and that $K$ is dense in $K_v.$ Suppose that we have a finite subextension $K_v \subset M \subset L_w.$ Is it then true that $M \cap L$ is dense in $M?$
This is certainly true if $M=K_v$ or $L_w,$ but the general case seems a bit trickier.