Flexible joints are widely used in ‘soft’ touching and holding, and they represent the main component of ultra-short radius drilling tools. The analysis of contact and motion characteristics is an essential issue in the design and development stage of flexible joints. In this study, a collision dynamics model of a ball cage flexible joint (BCFJ), which is suitable for the characteristics of small clearance and large load, is established. The model contains a nonlinear stiffness coefficient and can describe the contact force between the ball key and the raceways. Moreover, the computational procedure for the dynamic analysis of BCFJ with clearance is established, and the dynamic simulation for collision and contact between ball, cage, outer race, and inner race was carried out. By numerical calculation, the variation of contact force on the five contact points of the ball key and ball cage is discussed, and the influence of ball cage clearance on contact force between ball key, ball seat, and ball cage is obtained. The results indicate that the effects of the ball cage clearance on the contact force cannot be ignored, which is the main cause for the vibration of the flexible joint system, and the amplitude of the contact force will gradually increase with the increase of the clearance. The proposed model and procedure can analyze the dynamic behavior of flexible joints with small clearance and large load, providing a basis for further research on wear prediction and safety evaluation of the BCFJ with clearance.