The Majorana Demonstrator is an ultralow-background experiment searching for neutrinoless double-beta decay in ^{76}Ge. The heavily shielded array of germanium detectors, placed nearly a mile underground at the Sanford Underground Research Facility in Lead, South Dakota, also allows searches for new exotic physics. Free, relativistic, lightly ionizing particles with an electrical charge less than e are forbidden by the standard model but predicted by some of its extensions. If such particles exist, they might be detected in the Majorana Demonstrator by searching for multiple-detector events with individual-detector energy depositions down to 1 keV. This search is background-free, and no candidate events have been found in 285 days of data taking. New direct-detection limits are set for the flux of lightly ionizing particles for charges as low as e/1000.