Can dark matter - electron scattering explain the DAMA annual modulation signal?

The annually modulating $\sim$ keV scintillations observed in the DAMA/NaI and DAMA/Libra experiments might be due to dark matter - electron scattering. Such an explanation is now favoured given the stringent constraints on nuclear recoil rates obtained by LUX, SuperCDMS and other experiments. We suggest that multi-component dark matter models featuring light dark matter particles of mass $\sim$ MeV can potentially explain the data. A specific example, kinetically mixed mirror dark matter, is shown to have the right broad properties to consistently explain the experiments via dark matter - electron scattering. If this is the explanation of the annual modulation signal found in the DAMA experiments then a sidereal diurnal modulation signal is also anticipated. We point out that the data from the DAMA experiments show a diurnal variation at around 2.3$\sigma$ C.L. with phase consistent with that expected. This electron scattering interpretation of the DAMA experiments can potentially be probed in large xenon experiments (LUX, XENON1T,...), as well as in low threshold experiments (CoGeNT, CDEX, C4, ...) by searching for annually and diurnally modulated electron recoils.