Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0$.
By the classical Sobczyk Theorem, each separable Banach space is co-Sobczyk. But the class of co-Sobczyk spaces includes many non-separable Banach spaces. In particular, a Banach space $X$ is co-Sobczyk if each separable subspace of $X$ is contained in a complemented separable subspace. So, all classical Banach spaces $c_0(\Gamma)$ and $\ell_p(\Gamma)$ for $1\le p<\infty$, are co-Sobczyk for any set $\Gamma$.
I have a strong feeling that co-Sobczyk spaces have been studied in the theory of non-separable Banach spaces, so asking the MO commubnity for a proper reference and an existing terminology (I suspect that co-Sobczyk spaces are called differently).