We present the first constraints on the spin-dependent, inelastic scattering cross section of Weakly Interacting Massive Particles (WIMPs) on nucleons from XENON100 data with an exposure of 7.64$times$10$^3$,kg,day. XENON100 is a dual-phase xenon time projection chamber with 62,kg of active mass, operated at the Laboratori Nazionali del Gran Sasso (LNGS) in Italy and designed to search for nuclear recoils from WIMP-nucleus interactions. Here we explore inelastic scattering, where a transition to a low-lying excited nuclear state of $^{129}$Xe is induced. The experimental signature is a nuclear recoil observed together with the prompt de-excitation photon. We see no evidence for such inelastic WIMP-$^{129}$Xe interactions. A profile likelihood analysis allows us to set a 90% C.L. upper limit on the inelastic, spin-dependent WIMP-nucleon cross section of $3.3 times 10^{-38}$,cm$^{2}$ at 100,GeV/c$^2$. This is the most constraining result to date, and sets the pathway for an analysis of this interaction channel in upcoming, larger dual-phase xenon detectors.
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