We present a Banach space
X
\mathfrak X
with a Schauder basis of length
ω
1
\omega _1
which is saturated by copies of
c
0
c_0
and such that for every closed decomposition of a closed subspace
X
=
X
0
⊕
X
1
X=X_0\oplus X_1
, either
X
0
X_0
or
X
1
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
X
\mathfrak X
have “few operators” in the sense that every bounded operator
T
:
X
→
X
T:X \rightarrow \mathfrak {X}
from a subspace
X
X
of
X
\mathfrak {X}
into
X
\mathfrak {X}
is the sum of a multiple of the inclusion and a
ω
1
\omega _1
-singular operator, i.e., an operator
S
S
which is not an isomorphism on any non-separable subspace of
X
X
. We also show that while
X
\mathfrak {X}
is not distortable (being
c
0
c_0
-saturated), it is arbitrarily
ω
1
\omega _{1}
-distortable in the sense that for every
λ
>
1
\lambda >1
there is an equivalent norm
‖
|
⋅
‖
|
\| |\cdot \| |
on
X
\mathfrak {X}
such that for every non-separable subspace
X
X
of
X
\mathfrak {X}
there exist
x
,
y
∈
S
X
x,y\in S_X
such that
‖
|
x
‖
|
/
‖
|
y
‖
|
≥
λ
\| |x\| |/\| |y\| |\ge \lambda
.