I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.
You could try Karol Borsuk's Theory of Retracts. There is extensive discussion of ANRs. Further related conditions can be found in shape theory (work in the 1970s by Edwards and Geoghegan) but that requires a knowledge of shape theoretic ideas. (That is why I asked what sort of conditions you were looking for as this is taking you away from the central area of Banach spaces).