In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.