Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below?

If $X^{*}$ is $w^*$-separable, then for every Banach space $Y$, there exists an injective compact operator $T: X\to Y$ (see Goldberg and A.H. Kruse, The Existence of Compact Linear Maps Between Banach Spaces. Proc. A.M.S. 13 (1962), 808-811) and we know that a compact operator is not bounded below.

We also know that for $X$ nonseparable we may not find any injective compact operator into any Banach space (Existence of injective compact operators).