  The nature of dark matter (DM) is still under intense debate. Sub-galactic
scales are particularly critical, as different, currently viable DM models make
diverse predictions on the expected abundance and density profile of DM haloes
on these scales. We investigate the ability of sub-galactic DM haloes to act as
strong lenses on background compact sources, producing gravitational lensing
events on milli-arcsecond scales (milli-lenses), for different DM models. For
each DM scenario, we explore whether a sample of $\sim$ 5000 distant sources is
sufficient to detect at least one milli-lens. We develop a semi-analytical
model to estimate the milli-lensing optical depth as a function of the source's
redshift for various DM models. We employ the Press-Schechter formalism, as
well as results from recent N-body simulations to compute the halo mass
function, taking into account the appropriate spherically averaged density
profile of haloes for each DM model. We treat the lensing system as a
point-mass lens and invoke the effective surface mass density threshold to
calculate the fraction of a halo that acts as a gravitational lens. We study
three classes of dark matter models: cold DM, warm DM, and self-interacting DM.
We find that haloes consisting of warm DM turn out to be optically thin for
strong gravitational milli-lensing (zero expected lensing events). CDM haloes
may produce lensing events depending on the steepness of the concentration-mass
relation. Self-interacting DM haloes can efficiently act as gravitational
milli-lenses only if haloes experience gravothermal collapse, resulting in
highly dense central cores.
