Every separable Banach space is bi-Lipschitz isomorphic to a subset of $c_0$. But here are Banach spaces that do not uniformly embed in $c_0(\Gamma)$ for any $\Gamma$. See

• Jan Pelant, Petr Holický, Ondřej F. K. Kalenda, $C(K)$ spaces which cannot be uniformly embedded into $c_0(\Gamma)$, Fundamenta Mathemematicae 192 (2006) pp. 245–254 (journal abstract page).

Every separable Banach space is bi-Lipschitz isomorphic to a subset of $c_0$. But there are Banach spaces that do not uniformly embed in $c_0(\Gamma)$ for any $\Gamma$. See

• Jan Pelant, Petr Holický, Ondřej F. K. Kalenda, $C(K)$ spaces which cannot be uniformly embedded into $c_0(\Gamma)$, Fundamenta Mathematicae 192 (2006) pp. 245–254 (journal abstract page).

Every separable Banach space is bi-Lipschitz isomorphic to a subset of $c_0$. But there are Banach spaces that do not uniformly embed in $c_0(\Gamma)$ for any $\Gamma$. See Pelant et al, Fundamenta Mathemematicae 192 (2006) 245--254.

Every separable Banach space is bi-Lipschitz isomorphic to a subset of $c_0$. But here are Banach spaces that do not uniformly embed in $c_0(\Gamma)$ for any $\Gamma$. See

• Jan Pelant, Petr Holický, Ondřej F. K. Kalenda, $C(K)$ spaces which cannot be uniformly embedded into $c_0(\Gamma)$, Fundamenta Mathemematicae 192 (2006) pp. 245–254 (journal abstract page).