We report the results of a first experimental search for lepton number violation by four units in the neutrinoless quadruple-$beta$ decay of $^{150}$Nd using a total exposure of $0.19$ kg$cdot$y recorded with the NEMO-3 detector at the Modane Underground Laboratory (LSM). We find no evidence of this decay and set lower limits on the half-life in the range $T_{1/2}>(1.1-3.2)times10^{21}$ y at the $90%$ CL, depending on the model used for the kinematic distributions of the emitted electrons.
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