The wide-area damping control (WADC) designed by conventional model-based methods faces the challenge from the strong nonlinearity, time-varying structure, and disturbances from complex operating conditions of the interconnected multi-machine power systems. In this paper, as one of the data-driven methods, the model-free adaptive control (MFAC), which employs the pseudo-partial derivative (PPD) to linearize the nonlinear power system dynamically, is adopted to design WADC for its advantages of the low cost, simple structure & low calculation burden, and the analyzable stability. However, when applied to the WADC design, the prototype MFAC is infeasible due to the assumed known control direction, the non-zero steady-state controller output, and unconsidered disturbances. To solve these problems, a novel data-driven adaptive WADC, consisting of control direction correction, PPD updating, one-step-ahead weighted predictive control, is proposed, moreover, the stability of the closed-loop system considering disturbances is thoroughly analyzed. Finally, case studies with various disturbances are conducted to validate the proposed control strategy.