Bright radar reflections observed beneath the south polar layered deposits (SPLD) by the Mars Advanced Radar for Subsurface and Ionosphere Sounding instrument were interpreted to represent liquid water, but the required amounts of salt and heat to form and maintain liquids in this location are implausible given what is known about Mars. Here, we present another hypothesis that accounts for the bright reflections: hydrated and cold clay-rich deposits at the base of the SPLD create the observed radar response. To support this hypothesis, we present experimental measurements and wave propagation modeling that show that smectites, cooled to 230 K, have real and imaginary parts of the dielectric permittivity large enough to cause the bright reflections, even when mixed with other materials. Further, we find that absorptions attributable to these minerals are present in south polar orbital visible-near infrared reflectance spectra. Because these minerals are present at the south pole and can cause the reflections, we believe this to be a more viable scenario than the liquid water interpretation.