On bilipschitz extensions in real Banach spaces

Suppose that $E$ and $E^{\mathrm{\prime }}$ denote real Banach spaces with dimension at least 2, that $D≠E$ and $D^{\mathrm{\prime }}≠E^{\mathrm{\prime }}$ are bounded domains with connected boundaries, that $f:D\to D^{\mathrm{\prime }}$ is an $M$-QH homeomorphism, and that $D^{\mathrm{\prime }}$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism \bar \bar{D}\to \bar{D}' and $\bar{f}\mathrm{\mid }\mathrm{\partial }D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of V"ais"al"a is affirmative under an natural additional condition.