  We report on four radio-detected cosmic-ray (CR) or CR-like events observed
with the Antarctic Impulsive Transient Antenna (ANITA), a NASA-sponsored
long-duration balloon payload. Two of the four were previously identified as
stratospheric CR air showers during the ANITA-I flight. A third stratospheric
CR was detected during the ANITA-II flight. Here we report on characteristics
these three unusual CR events, which develop nearly horizontally, 20-30~km
above the surface of the Earth. In addition, we report on a fourth steeply
upward-pointing ANITA-I CR-like radio event which has characteristics
consistent with a primary that emerged from the surface of the ice. This
suggests a possible $\tau$-lepton decay as the origin of this event, but such
an interpretation would require significant suppression of the Standard Model
${\tau}$-neutrino cross section.
