  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.
