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The feedback between dyke and sill intrusions and the evolution of stresses within volcanic systems is poorly understood, despite its importance for magma transport and volcano instability. Long-lived ocean island volcanoes are crosscut by thousands of dykes, which must be accommodated through a combination of flank slip and visco-elastic deformation. Flank slip is dominant in some volcanoes (e.g., Kilauea), but how intrusions are accommodated in other volcanic systems remains unknown. Here we apply digital mapping techniques to collect > 400,000 orientation and aperture measurements from 519 sheet intrusions within Volcán Taburiente (La Palma, Canary Islands, Spain) and investigate their emplacement and accommodation. We show that vertically ascending dykes were deflected to propagate laterally as they approached the surface of the volcano, forming a radial dyke swarm, and propose a visco-elastic model for their accommodation. Our model reproduces the measured dyke-aperture distribution and predicts that stress accumulates within densely intruded regions of the volcano, blocking subsequent dykes and causing eruptive activity to migrate. These results have significant implications for the organisation of magma transport within volcanic edifices, and the evolution and stability of long-lived volcanic systems.