Using neutrinos produced at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory (ORNL), the COHERENT collaboration has studied the $\mathrm{Pb}({\ensuremath{\nu}}_{e},\mathrm{X}n)$ process with a lead neutrino-induced-neutron (NIN) detector. Data from this detector are fit jointly with previously collected COHERENT data on this process. A combined analysis of the two datasets yields a cross section that is ${0.29}_{\ensuremath{-}0.16}^{+0.17}$ times that predicted by the MARLEY event generator using experimentally-measured Gamow-Teller strength distributions, consistent with no NIN events at $1.8\ensuremath{\sigma}$. This is the first inelastic neutrino-nucleus process COHERENT has studied, among several planned exploiting the high flux of low-energy neutrinos produced at the SNS.