Let $B$ be an integral Banach ring, meaning it is complete with respect to a norm and it is an integral domain. Let $F$ be its fraction field. Let $\widehat{F}$ be the completion of $F$ with respect to the norm on $A$.
Is there an Artin approximation theorem for $F$ and $\widehat{F}$? Meaning solutions of equations over $\widehat{F}$ imply solutions of equations over $F$?
If it is not true in general, are there cases where it is true?
