We measure the frequencies and damping times for the radial breathing mode of a trapped, strongly interacting Fermi gas of lithium-6 atoms. A 50-50 mixture of the two lowest spin states is cooled to temperatures ranging from 1.2 TF to 0.15 TF at magnetic fields near a Feshbach resonance. At the lowest temperatures, Pauli blocking is expected to suppress the collision rate. Nevertheless, just above resonance, the measured frequency is in very good agreement with predictions for a hydrodynamic Fermi gas in the unitarity limit and the damping time increases strongly as the temperature is lowered. The results are inconsistent with predictions for both the collisionless mean field regime and for collisional hydrodynamics. These observations provide the first evidence for superfluid hydrodynamics in a strongly attractive Fermi gas.