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The kinetics of polarization switching in ultrathin ferroelectric polymer films exhibit a critical behavior; there is a pronounced slowing just above the coercive field and just below the critical temperature. The critical slowing is observed in the switching kinetics of ferroelectric Langmuir-Blodgett films of 70% vinylidene-fluoride and 30% trifluoroethylene copolymer with thickness up to 30 ML. Thicker films exhibit an exponential dependence on the field and temperature that is normally associated with extrinsic switching by domain nucleation and growth. We show that the critical behavior arises naturally from the dynamics of homogeneous intrinsic switching in the context of mean-field theory. These results have important implications for the modeling and control of the ferroelectric films in nonvolatile computer memories and solid-state data storage media.