In this paper, using theorems on the continuous dependence of the solution of differential inclusions on the perturbation, we obtain high-order exact penalty theorems for nonconvex extremal problems of differential inclusions in the space of Banach-valued absolutely continuous functions. Using the type of the distance function in the classes of 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz functions, the nonconvex extremal problem for differential inclusions is reduced to a variational problem and the necessary condition for the extremum of a high-order is obtained. The paper also shows that the used functions type of distance function satisfy the 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz conditions.