Sobczyk's theorem: if $Z$ is a subspace of a separable Banach space $X$ that is isomorphic to $c_0$, then $Z$ is complemented in $X$, fails for many non-separable spaces such as $X=\ell_\infty\cong C( \beta \mathbb N)$.

For related reasons, the Borsuk–Dugundji extension theorem (which says that if $F$ is a closed, metrisable subspace of a compact space $K$, then you can apply the Tietze–Urysohn extension theorem in a linear way, i.e., there is a contractive operator $T\colon C(F)\to C(K)$ such that $(Tf)|_F=f$ for $f\in C(F)$) fails when $F$ is non-metrisable, that is, $C(F)$ is non-separable in the uniform norm. (Here, $F = \beta \mathbb N\setminus \mathbb N \subset \beta\mathbb N$ is the easiest counterexample.)