Given Banach spaces $X$, $Y$ and a bounded operator $T:X\to Y$ with non-closed range, a perturbation argument shows that there exists an infinite-dimensional closed subspace $M$ of $X$ such that the restriction of $T$ to $M$ is compact.

Let $J:L_\infty(0,1)\to L_1(0,1)$ denote the natural inclusion. Is it posible to find a non-separable closed subspace $N$ of $L_\infty(0,1)$ such that the restriction of $J$ to $N$ is compact?