The polarization switching kinetics were measured at the nanoscale in continuous thin films of a ferroelectric copolymer of vinylidene fluoride and trifluoroethylene. The dependence of the switching rate on voltage for a 54-nm thick film exhibits extrinsic nucleation and domain-growth type kinetics with no true threshold coercive field, and is qualitatively different from the behavior of an 18-nm thick film, which exhibits intrinsic switching kinetics, and a true threshold field. The results are consistent with studies of thin film capacitors of much larger area and with a recent refinement of the theory of the critical size for intrinsic switching.