An implicit discrete unified gas kinetic scheme (DUGKS) is developed for multiscale steady flows of binary gas mixtures by solving the Andries-Aoki-Perthame kinetic model (AAP). To ensure the high convergence efficiency for all flow regimes, the microscopic and macroscopic asynchronous iterative strategies are used, where both the macroscopic and microscopic equations are solved iteratively by the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method. The macroscopic iteration is conducted to solve the macroscopic governing equations containing source terms as an implicit prediction step to evaluate the local equilibrium state of the microscopic evolution, and the macroscopic flux used in the macroscopic iteration is obtained by taking moments of the distribution function. Besides, to keep the asymptotic preserving properties, the numerical flux across the cell interface is reconstructed by the characteristic solution of the kinetic governing equations for both species like the explicit DUGKS for a single gas. Several numerical tests, including the Couette flow, the lid-driven cavity flow, and the flows through a slit of different mixtures, are simulated to verify the accuracy and efficiency of the present scheme for binary mixtures. Furthermore, compared to the explicit DUGKS, the implicit scheme improves the computational efficiency by 1-2 orders of magnitude.