A $c_0$-saturated Banach space with no long unconditional basic sequences

We present a Banach space $\mathfrak{X}$ with a Schauder basis of length $\omega_1$ which is saturated by copies of $c_0$ and such that for every closed decomposition of a closed subspace $X=X_0\oplus X_1$, either $X_0$ or $X_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak{X}$ have ``few operators'' in the sense that every bounded operator $T:X \rightarrow \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $\omega_1$-singular operator, i.e., an operator $S$ which is not an isomorphism on any non-separable subspace of $X$. We also show that while $\mathfrak{X}$ is not distortable (being $c_0$-saturated), it is arbitrarily $\omega_{1}$-distortable in the sense that for every $\lambda>1$ there is an equivalent norm $\Vert \vert\cdot\Vert \vert$ on $\mathfrak{X}$ such that for every non-separable subspace $X$ of $\mathfrak{X}$ there exist $x,y\in S_X$ such that $\Vert \vert x\Vert \vert/\Vert \vert y\Vert \vert\ge \lambda$.