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A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ isconstructed that is the first known example of a $\mathscr{L}_\infty$-space notcontaining $c_0$, $\ell_1$, or reflexive subspaces and answers a question posedby J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfiesthe "scalar-plus-compact" property and it is the first known space withoutreflexive subspaces having this property. It is constructed using theBourgain-Delbaen method in combination with a recent version of saturationunder constraints in a mixed-Tsirelson setting. As a result, the space$\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decompositionand does not contain a boundedly complete sequence.

The scalar-plus-compact property in spaces without reflexive subspaces