  We study the effects of anomalous thresholds on the non-local form factors
describing the hadronization of the light-quark contribution to
$B\to(P,V)\gamma^*$ transitions. Starting from a comprehensive discussion of
anomalous thresholds in the triangle loop function for different mass
configurations, we detail how the dispersion relation for $\pi\pi$ intermediate
states is affected by contour deformations mandated by the anomalous branch
points. Phenomenological estimates of the size of the anomalous contributions
to the form factors are provided with couplings determined from measured
branching fractions and Dalitz plot distributions. Our key finding is that
anomalous effects are suppressed on the $\rho(770)$ resonance, while off-peak
the effects can become as large as $\mathcal{O}(10\%)$ of the full
(light-quark-loop induced) non-local form factors. We comment on future
generalizations towards higher intermediate states and the charm loop,
outlining how the dispersive framework established in this work could help
improve the non-local form factors needed as input for a robust interpretation
of $B\to (P,V)\ell^+\ell^-$ decays.
