We address the problem of UV/IR mixing in noncommutative
quantum field theories from the perspective of braided $L_\infty$-structures and the Batalin--Vilkovisky formalism. We describe the example of
braided noncommutative scalar
field theory and its quantization using braided homological perturbation theory. The formalism is illustrated through one-loop calculations of the
two-point functions for $\phi^4$-theory in four dimensions and $\phi^3$-theory in six dimensions. In both cases we find that there are no
non-planar diagrams and no UV/IR mixing.