  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.
