  We prove that the Grothendieck ring of the monoidal category of finitely
generated graded projective modules for the nil-Brauer category is isomorphic
to an integral form of the split $\imath$-quantum group of rank one. Under this
isomorphism, the indecomposable graded projective modules correspond to the
$\imath$-canonical basis. We also introduce a new PBW basis for the
$\imath$-quantum group and show that it is categorified by standard modules for
the nil-Brauer category. Finally, we derive character formulae for irreducible
graded modules and deduce various branching rules.
