The author uses the secant method for solving a nonlinear operator equation in a Banach space setting. Assuming that the derivative of the governing operator is Hölder continuous and that the initial divided difference operator is invertible, he proves that the sequence produced by the secant iteration method converges to a unique solution of the operator equation. Moreover, he gives an error estimate and presents two examples as possible applications of the theory, which do not satisfy the assumptions of the Newton-Kantorovich method.