We report on an upward traveling, radio-detected cosmic-ray-like impulsive event with characteristics closely matching an extensive air shower. This event, observed in the third flight of the Antarctic Impulsive Transient Antenna (ANITA), a NASA-sponsored long-duration balloon payload, is consistent with a similar event reported in a previous flight. These events could be produced by the atmospheric decay of an upward-propagating τ lepton produced by a ν_{τ} interaction, although their relatively steep arrival angles create tension with the standard model neutrino cross section. Each of the two events have a posteriori background estimates of ≲10^{-2} events. If these are generated by τ-lepton decay, then either the charged-current ν_{τ} cross section is suppressed at EeV energies, or the events arise at moments when the peak flux of a transient neutrino source was much larger than the typical expected cosmogenic background neutrinos.