The feasibility of the generation of bright ultrashort gamma-ray pulses is demonstrated in the interaction of a relativistic electron bunch with a counterpropagating tightly focused superstrong laser beam in the radiation-dominated regime. The Compton scattering spectra of gamma radiation are investigated using a semiclassical description for the electron dynamics in the laser field and a quantum electrodynamical description for the photon emission. We demonstrate the feasibility of ultrashort gamma-ray bursts of hundreds of attoseconds and of dozens of megaelectronvolt photon energies in the near-backwards direction of the initial electron motion. The tightly focused laser field structure and the radiation reaction are shown to be responsible for such short gamma-ray bursts, which are independent of the durations of the electron bunch and of the laser pulse. The results are measurable with the laser technology available in the near future.