The study of numerical methods for stiff functional differential equations is mostly carried out in the inner product space based on the assumption that the one-sided Lipschitz constant has a moderate size, whereas for some stiff problems, the one-sided Lipschitz constant inevitably takes very large positive values. Therefore, it is necessary to break through the limitations of inner product space and one-sided Lipschitz constant and study the related numerical methods directly in the Banach space. For the nonlinear composite stiff Volterra functional differential equation in Banach space, the non-stiff part is solved by the explicit Euler method and the stiff part is solved by the implicit Euler method, obtained from which is the implicit-explicit Euler method for solving the problem. The stability and asymptotic stability about the proposed method are established, and the numerical results verify the correctness of the obtained theory.