I am interested in examples of Banach spaces $X$ satisfying the following two conditions:

(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.

(2) $X$ is isomorphic to its hyperplanes.

Observations:

(a) A space satisfying these two conditions must be non-separable [GK], and it cannot admit an infinite dimensional separable complemented subspace.

(b) There are non-separable H.I. spaces that satisfy (1), but H.I. spaces fail (2).

(b) In [AKo] we can find an example of a non-separable $C(K)$ space in which every injective operator is surjective, but it fails (2).