La Palma island is one of the highest potential risks in the volcanic archipelago of the Canaries and therefore it is important to carry out an in-depth study to define its state of unrest. This has been accomplished through the use of satellite radar observations and an original state-of-the-art interpretation technique. Here we show the detection of the onset of volcanic unrest on La Palma island, most likely decades before a potential eruption. We study its current evolution seeing the spatial and temporal changing nature of activity at this potentially dangerous volcano at unprecedented spatial resolutions and long time scales, providing insights into the dynamic nature of the associated volcanic hazard. The geodetic techniques employed here allow tracking of the fluid migration induced by magma injection at depth and identifying the existence of dislocation sources below Cumbre Vieja volcano which could be associated with a future flank failure. Therefore they should continue being monitored using these and other techniques. The results have implications for the monitoring of steep-sided volcanoes at oceanic islands.