  The observed amount of lithium for low metallicity population II stars (known
as the Spite plateau) is a factor of $\sim 3-5$ lower than the predictions of
the standard cosmology. Since the observations are limited to the local
Universe (halo stars, globular clusters and satellites of the Milky Way) it is
possible that certain physical processes may have led to the spatial separation
of lithium and local reduction of [Li/H]. We study the question of lithium
diffusion after the cosmological recombination in sub-Jeans dark matter haloes,
taking into account that more than 95% of lithium remains in the singly-ionized
state at all times. Large scattering cross sections on the rest of the ionized
gas leads to strong coupling of lithium to protons and its initial direction of
diffusion coincides with that of H$^+$. In the rest frame of the neutral gas
this leads to the diffusion of H$^+$ and Li$^+$ out of overdensities with the
trend of reducing [Li/H] in the minima of gravitational wells relative to the
primordial value. We quantify this process and argue that, with certain
qualifications, it may have played a significant role in creating local lithium
deficiency within the primordial dark matter haloes, comparable to those
observed along the Spite plateau.
