In the present work we examine the possibility of detecting electrons in light dark matter searches. These detectors are considered to be the most appropriate for detecting dark matter particles with a mass in the MeV region. We analyze theoretically some key issues involved in such detection. More specifically we consider a particle model involving WIMPs interacting with fermions via $Z$-exchange. We find that for WIMPs with mass in the electron mass range the cross section for WIMP-atomic electron scattering is affected by the electron binding. For WIMPs more than 20 times heavier than the electron, the binding affects the kinematics very little. As a result, many electrons can be ejected with energy which increases linearly with the WIMP mass, but the cross section is somewhat reduced depending on the bound state wave function employed. On the other hand for lighter WIMPs, the effect of binding is dramatic. More specifically at most 10 electrons, namely, those with binding energy below 10 eV, become available even in the case of WIMPs with a mass as large as 20 times the electron mass. Even fewer electrons contribute if the WIMPs are lighter. The cross section is, however, substantially enhanced by the Fermi function corrections, which become more important at low energies of the outgoing electrons. Thus events of 0.5–2.5 per kg-y become possible.