Kalton's Problem 2: does a bi-Lipschitz copy of $c_0$ force a linear copy of $c_0$?

Theorem 1.1: there is a separable real Banach space $Z$ with no closed subspace isomorphic to $c_0$ and an onto bi-Lipschitz map $F:Z\oplus_\infty c_0\to Z$. Restricting $F$ to the $c_0$ summand embeds $c_0$ bi-Lipschitzly in $Z$, so Problem 2 has a negative answer. With Aharoni's theorem, $Z$ is bi-Lipschitz universal for separable metric spaces, and $Z$, $Z\oplus_\infty c_0$ form a second separable Lipschitz-isomorphic, linearly non-isomorphic pair. Real scalars only; it does not address analogous questions for other classical spaces.

Theorem 1.1: there is a separable real Banach space $Z$ with no closed subspace isomorphic to $c_0$ and an onto bi-Lipschitz map $F:Z\oplus_\infty c_0\to Z$. Restricting $F$ to the $c_0$ summand embeds $c_0$ bi-Lipschitzly in $Z$, so Problem 2 has a negative answer. With Aharoni's theorem, $Z$ is bi-Lipschitz universal for separable metric spaces, and $Z$, $Z\oplus_\infty c_0$ form a second separable Lipschitz-isomorphic, linearly non-isomorphic pair. Real scalars only; it does not address analogous questions for other classical spaces.