<p>This article presents the local convergence analysis of a fifth-order Potra–Pták-like iteration method for solving nonlinear equations in Banach spaces. Our convergence analysis depends exclusively on the first-order derivative, in contrast to existing results that establish fifth-order convergence using the classical Taylor series approach, which requires higher-order derivatives. A comprehensive extended ball convergence theorem for the proposed method is developed, providing explicit expressions for the radius of convergence, error estimates, and the uniqueness domain of the solution. Furthermore, the dynamic behavior of this iterative method is explored using the concept of basins of attraction, which visually demonstrate the convergence regions for different complex polynomial equations. We analyze and compare two cases of the method, known as PTKM⁵-I and PTKM⁵-II, in terms of their numerical examples and dynamic characteristics. The numerical results are also compared with the existing iteration method, showing that both cases of the proposed method have improved performance compared to the existing method, with PTKM⁵-I providing a wider convergence region than PTKM⁵-II. Theoretical and numerical analyses confirm the effectiveness and robustness of the proposed iteration scheme.</p>