The technique of tracking a non-cooperative manoeuvring satellite is important for Space Situation Awareness (SSA). However, the classical extended Kalman filter cannot work successfully in this situation. Motivated by this problem, a novel Extended Kalman Filter with Input Detection and Estimation (EKF/IDE) method is proposed in this paper for tracking a non-cooperative satellite with impulsive manoeuvres. The impulsive manoeuvre is modelled as an unknown acceleration without any prior information. An unbiased minimum-variance input and state estimation method is introduced to estimate the manoeuvre acceleration. An approach based on the Mahalanobis distance of the manoeuvre estimate error is proposed for manoeuvre detection. With the impulsive manoeuvre being detected and estimated accurately, an adaptive extended Kalman filter is proposed to estimate the state of the target. An approach of covariance inflation is proposed to deal with the manoeuvre during the unobserved period. Simulations and Monte Carlo experiments are implemented to demonstrate the feasibility and validity of the proposed method. The results of simulations show that the proposed method can accurately detect and estimate unknown impulsive manoeuvres of a non-cooperative satellite. Through the compensation of the estimated manoeuvre, the estimation of position and velocity after the manoeuvre maintains the accuracy of the pre-manoeuvre period, demonstrating the robustness of the proposed method against unknown manoeuvres.