  A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is
constructed that is the first known example of a $\mathscr{L}_\infty$-space not
containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed
by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies
the "scalar-plus-compact" property and it is the first known space without
reflexive subspaces having this property. It is constructed using the
Bourgain-Delbaen method in combination with a recent version of saturation
under constraints in a mixed-Tsirelson setting. As a result, the space
$\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decomposition
and does not contain a boundedly complete sequence.
