By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.


Is this result known to fail for nonseparable spaces? That is, is there a known example of two (necessarily nonseparable) Banach spaces $X,Y$ such that $X$ embeds isometrically into $Y$, but such that there is no linear isometric embedding of $X$ into $Y$?


This question was previously asked on [MSE](https://math.stackexchange.com/questions/4192857/what-is-an-example-of-two-banach-spaces-x-y-such-that-x-embeds-isometrically) but received no answer there.



What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$?
Yes, if $H$ is a nonseparable Hilbert space then it embeds isometrically into the Arens-Eells space ${\rm AE}(H)$, but not linearly isometrically, or even linearly homeomorphically. See Theorem 5.21 of my book *Lipschitz Algebras* (second edition).


As I explain in the notes to that chapter, a more general version of this statement was claimed in a paper of Godefroy and Kalton, but their proof is erroneous and, as far as I can tell, not fixable. However, some of the ideas of my argument are based on theirs.


