Exciton-polaritons offer a versatile platform for realization of all-optical integrated logic gates due to the strong effective optical nonlinearity resulting from the exciton-exciton interactions. In most of the current excitonic materials there exists a direct connection between the exciton robustness to thermal fluctuations and the strength of exciton-exciton interaction, making materials with highest levels of exciton nonlinearity applicable at cryogenic temperatures only. Here, we show that strong polaronic effects, characteristic for perovskite materials, allow to overcome this limitation. Namely, we demonstrate the record-high value of the nonlinear optical response in nanostructured organic-inorganic halide perovskite MAPbI$_3$, experimentally detected as 19.7 meV blueshift of the polariton branch under femtosecond laser irradiation. This is substantially higher than characteristic values for the samples based on conventional semiconductors and monolayers of transition metal dichalcogenides. The observed strong polaron-enhanced nonlinearity exists for both tetragonal and orthorombic phases of MAPbI$_3$, and remains stable at elevated temperatures.
