  We give a complete solution to the Borel-Ritt problem in non-uniform spaces
$\mathscr{A}^-_{(M)}(S)$ of ultraholomorphic functions of Beurling type, where
$S$ is an unbounded sector of the Riemann surface of the logarithm and $M$ is a
strongly regular weight sequence. Namely, we characterize the surjectivity and
the existence of a continuous linear right inverse of the asymptotic Borel map
on $\mathscr{A}^-_{(M)}(S)$ in terms of the aperture of the sector $S$ and the
weight sequence $M$. Our work improves previous results by Thilliez [10] and
Schmets and Valdivia [9].
