<p>We propose a new Lie-algebraic approach to the construction of Lax–Sato integrable dispersion-free systems on functional supermanifolds by means of a centrally extended semidirect sum of the loop Lie algebra of superconformal vector fields on a supercircle and its regular dual space based on the general Adler–Kostant–Symes Lie-algebraic scheme. By using this approach, we obtain the Lax–Sato integrable superanalogs for some systems of Mikhalev–Pavlov-type dispersion-free equations given on functional supermanifolds of four commuting and numerous anticommuting independent variables and find the left gradients of the Casimir invariant reduced to the orbits of coadjoint action of the central extension related to these systems, as well as the associated pairs of compatible Poisson operators.</p>