EDIT: since I'm not an expert on Banach spaces, I feel I shouldn't say anything more, but anyway; an essential ingredient is an **exact** formula in Hilbert spaces for $\|x + y\|^2$. *Just an idea* (*perhaps I am being stupid*): maybe if you have a Banach space where $\| x + y \| = F(\|x\|, \|y \|, g(x,y))$ for some reasonably simple functions $F$, $g$ then something can be said.

If you examine the proof for Hilbert spaces, it makes essential use of the scalar product; so it's not really surprising that it doesn't work for general Banach spaces. The norm is a nice enough structure to do a lot of things, but not *that* nice.

It also demonstrates that Banach spaces have far more detailed structure than just ordinary vector spaces, but that Hilbert spaces have even more structure again. In fact, there are many properties Hilbert spaces have which general Banach spaces don't (and many which even characterise Hilbert space uniquely).