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Nonparametric regression on complex domains has been a challenging task as most existing methods, such as ensemble models based on binary decision trees, are not designed to account for intrinsic geometries and domain boundaries. This article proposes a Bayesian additive regression spanning trees (BAST) model for nonparametric regression on manifolds, with an emphasis on complex constrained domains or irregularly shaped spaces embedded in Euclidean spaces. Our model is built upon a random spanning tree manifold partition model as each weak learner, which is capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints. Utilizing many nice properties of spanning tree structures, we design an efficient Bayesian inference algorithm. Equipped with a soft prediction scheme, BAST is demonstrated to significantly outperform other competing methods in simulation experiments and in an application to the chlorophyll data in Aral Sea, due to its strong local adaptivity to different levels of smoothness.