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Dielectric elastomer generators (DEGs) are a class of capacitive solid-state devices that employ highly stretchable dielectrics and conductors to convert mechanical energy into high-voltage direct-current electricity. Their promising performance in terms of convertible energy and power density has been mostly proven in quasi-static experimental tests with prescribed deformation. However, the assessment of their ability in harvesting energy from a dynamic oscillating source of mechanical energy is crucial to demonstrate their effectiveness in practical applications. This paper reports a first demonstration of a DEG system that is able to convert the oscillating energy carried by water waves into electricity. A DEG prototype is built using a commercial polyacrylate film (VHB 4905 by 3M) and an experimental campaign is conducted in a wave-flume facility, i.e. an artificial basin that makes it possible to generate programmed small-scale waves at different frequencies and amplitudes. In resonant conditions, the designed system demonstrates the delivery of a maximum of 0.87 W of electrical power output and 0.64 J energy generated per cycle, with corresponding densities per unit mass of dielectric elastomer of 197 W/kg and 145 J/kg. Additionally, a notable maximum fraction of 18% of the input wave energy is converted into electricity. The presented results provide a promising demonstration of the operation and effectiveness of ocean wave energy converters based on elastic capacitive generators.