The detection of fast variations of the tera-electron volt (TeV) (10(12) eV) gamma-ray flux, on time scales of days, from the nearby radio galaxy M87 is reported. These variations are about 10 times as fast as those observed in any other wave band and imply a very compact emission region with a dimension similar to the Schwarzschild radius of the central black hole. We thus can exclude several other sites and processes of the gamma-ray production. The observations confirm that TeV gamma rays are emitted by extragalactic sources other than blazars, where jets are not relativistically beamed toward the observer.