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Assuming that Dark Matter (DM) is made of fermions in the sub-GeV mass range with interactions dominated by electromagnetic moments of higher order, such as the electric and magnetic dipoles or the anapole moment, we show that direct detection experiments searching for atomic ionisation events in xenon targets can shed light on whether DM is a Dirac or Majorana particle. Specifically, we find that between about 45 (120) and 610 (1700) signal events are required to reject Majorana DM in favour of Dirac DM with a statistical significance corresponding to 3 (5) standard deviations. The exact number of DM signal events corresponding to a given significance depends on the relative size of the anapole, magnetic dipole and electric dipole contributions to the expected rate of DM-induced atomic ionisations under the Dirac hypothesis. Our conclusions are based on Monte Carlo simulations and the likelihood ratio test. While the use of asymptotic formulae for the latter is standard in many applications, here it requires a non-trivial extension to the case where one of the hypotheses lies on the boundary of the parameter space. Our results constitute a solid proof of concept about the possibility of using direct detection experiments to reject the Majorana DM hypothesis when the DM interactions are dominated by higher-order electromagnetic moments.