The Antarctic Impulsive Transient Antenna, a NASA long-duration balloon payload, searches for radio emission from interactions of ultrahigh-energy neutrinos in polar ice. The third flight of the Antarctic Impulsive Transient Antenna was launched in December 2014 and completed a 22-day flight. We present the results of three analyses searching for Askaryan radio emission of neutrino origin. In the most sensitive of the analyses, we find one event in the signal region on an expected background of 0.7-0.3+0.5. Though consistent with the background estimate, the event remains compatible with a neutrino hypothesis even after additional postunblinding scrutiny.