Strong evidence for self-excited emission of coherent synchrotron radiation in the microwave spectral region was observed at the Synchrotron Ultraviolet Radiation Facility (SURF III) electron storage ring at the NIST. The microwave emission between 25 and 35 mm was dominated by intense bursts of radiation. The intensity enhancement during these bursts was on the order of 10 000 to 50 000 over the incoherent value. The shape, width, and period of the bursts depend strongly on the operational parameters of the storage ring. Coherent microwave emission was observed only when the beam was unstable, namely, during bunch-length relaxation oscillations. We report on the measurements of the microwave bursts, and correlate the data with signals from a beam monitor electrode and photodiode detector. The coherent enhancement of the radiation intensity is ascribed to spontaneous self-induced microbunching of the electrons within the bunch.