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We present the first catalog of gamma-ray sources emitting above 56 and 100 TeV with data from the High Altitude Water Cherenkov Observatory, a wide field-of-view observatory capable of detecting gamma rays up to a few hundred TeV. Nine sources are observed above 56 TeV, all of which are likely galactic in origin. Three sources continue emitting past 100 TeV, making this the highest-energy gamma-ray source catalog to date. We report the integral flux of each of these objects. We also report spectra for three highest-energy sources and discuss the possibilit