Abstract This paper presents a quadratic-penalty type method for solving linearly-constrained composite nonconvex-concave min-max problems. The method consists of solving a sequence of penalty subproblems which, due to the min-max structure of the problem, are potentially nonsmooth but can be approximated by smooth composite nonconvex minimization problems. Each of these penalty subproblems is then solved by applying an accelerated inexact proximal point method to its corresponding smooth composite nonconvex approximation. Iteration complexity bounds for obtaining approximate stationary points of the linearly-constrained composite nonconvex-concave min-max problem are also established.