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