Elves are a class of transient luminous events, with a radial extent typically greater than 250 km, that occur in the lower ionosphere above strong electrical storms. We report the observation of 1,598 elves, from 2014 to 2016, recorded with unprecedented time resolution (100 ns) using the fluorescence detector (FD) of the Pierre Auger Cosmic-Ray Observatory. The Auger Observatory is located in the Mendoza province of Argentina with a viewing footprint for elve observations of 3.106 km2, reaching areas above the Pacific and Atlantic Oceans, as well as the Córdoba region, which is known for severe convective thunderstorms. Primarily designed for ultrahigh energy cosmic-ray observations, the Auger FD turns out to be very sensitive to the ultraviolet emission in elves. The detector features modified Schmidt optics with large apertures resulting in a field of view that spans the horizon, and year-round operation on dark nights with low moonlight background, when the local weather is favorable. The measured light profiles of 18% of the elve events have more than one peak, compatible with intracloud activity. Within the 3-year sample, 72% of the elves correlate with the far-field radiation measurements of the World Wide Lightning Location Network. The Auger Observatory plans to continue operations until at least 2025, including elve observations and analysis. To the best of our knowledge, this observatory is the only facility on Earth that measures elves with year-round operation and full horizon coverage.