We consider design-based causal inference in settings where randomized treatments have effects that bleed out into space in complex ways that overlap and in violation of the standard “no interference” assumption for many causal inference methods. We define a spatial “average marginalized response,” which characterizes how, in expectation, units of observation that are a specified distance from an intervention point are affected by treatments at that point, averaging over effects emanating from other intervention points. We establish conditions for non-parametric identification, asymptotic distributions of estimators, and recovery of structural effects. We propose methods for both sample-theoretic and permutation-based inference. We provide illustrations using randomized field experiments on forest conservation and health.