We present limits on the WIMP-nucleon cross section for inelastic dark matter derived from the 2008 run of ZEPLIN-III. Cuts, notably on scintillation pulse shape and scintillation-to-ionisation ratio, give a net exposure of 63 kg.days in the range 20-80keV nuclear recoil energy, in which 6 events are observed. Upper limits on signal rate are derived from the maximum empty patch in the data. Under standard halo assumptions a small region of parameter space consistent, at 99% CL, with causing the 1.17 ton.year DAMA modulation signal is allowed at 90% CL: it is in the mass range 45-60 GeV with a minimum CL of 88%, again derived from the maximum patch. This is the tightest constraint on that explanation of the DAMA result yet presented using a xenon target.
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