Do sufficiently large Banach spaces admit non-compact operators with not too large range?

As in the title,

does there exist a cardinal number $\lambda$ such that for every Banach space $X$ of density/cardinality at least $\lambda$ there exists a non-compact bounded, linear operator $T\colon X\to X$ such that the range $T[X]$ has smaller density than $X$?

By density I mean the minimal cardinality of a dense set. This question could be strengthened and we may ask about non-compact operators with separable range.

Some remarks are in order.

1. In general, one cannot expect $T$ to be idempotent (a projection), at least under GCH. (See this thread.)
2. Should the answer be positive, we would get for free examples of unital Banach algebras that are not isomorphic to Calkin algebras, that is, algebras of the form $B(X)/K(X)$, where $B(X)$ is the algebra of bounded operators on a space $X$ and $K(X)$ is the ideal of compact operators. (This suggests that it may be an open problem in the sense that it perhaps does not follow from a linear combination of known results.)

3. We lack examples of non-separable spaces $X$ for which this statement is false, that is potentially $\lambda$ could be $\omega_1$.