We report on the observation of spin whirls in a radially expanding polariton condensate formed under nonresonant optical excitation. Real space imaging of polarization- and time-resolved photoluminescence reveals a spiralling polarization pattern in the plane of the microcavity. Simulations of the spatiotemporal dynamics of a spinor condensate reveal the crucial role of polariton interactions with a spinor exciton reservoir. Harnessing spin-dependent interactions between the exciton reservoir and polariton condensates allows for the manipulation of spin currents and the realization of dynamic collective spin effects in solid-state systems.