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 but received no answer there.
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.
