The cross sections of neutron-induced nuclear reactions are commonly large at low energies. Given this feature, the 7Be(n, p)7Li process with slow 7Be is studied in connection with the cosmological lithium problem. An idea is tested whether slow (non-Maxwellian) 7Be born in the radiative capture 3He(α, γ)7Be reaction can boost the 7Be(n, p)7Li(p, α)α mechanism of 7Li burn-up in the early universe, reducing the amount of primordial 7Li. The rate of this sequential process is examined and its effect on the 7Li abundance is clarified. In the absence of 7Be thermalization in the primordial plasma, the above mechanism can decrease 7Li/H by several percent, leaving unchanged the abundances of D and 4He which are consistent with observations. However, taking into account the 7Be thermalization due to scattering off electrons, positrons, nuclei, and photons radically suppresses the effect. It is found that the typical thermalization time of 7Be is less than 10−5 s, while the fraction of non-Maxwellian particles in the 7Be component is at most 2 × 10−7. Under this condition, the corresponding impact on 7Li/H is estimated to be no more than 10−5%.