Collective behavior in many-body systems is the origin of many fascinating phenomena in nature, ranging from the formation of clouds to magnetic properties of solids. We report on the observation of collective spin dynamics in an ultracold Fermi sea with large spin. As a key result, we observed long-lived and large-amplitude coherent spin oscillations driven by local spin interactions. At ultralow temperatures, Pauli blocking stabilizes the collective behavior, and the Fermi sea behaves as a single entity in spin space. With increasing temperature, we observed a stronger damping associated with particle-hole excitations. Unexpectedly, we found a high-density regime where excited spin configurations are collisionally stabilized. Our results reveal the intriguing interplay between microscopic processes either stimulating or suppressing collective effects in a fermionic many-body system.