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A ball cage flexible drill pipe is a new type of ultra-short-radius drilling tool, which consists of multiple flexible joints hinged together. During the drilling process, the flexible members will come into contact and wear, which reduces the efficiency of load transfer. The multi-body collision contact dynamics model was proposed to study the performance of the ball cage flexible drill pipe. The method considered the influence of the borehole curvature. The kinematic equations of the ball cage flexible drill pipe were established. The Lankarani–Nikravesh collisional contact model was used to characterize the normal contact force, and the Coulomb friction model was used to describe the tangential contact force. The multi-body motion state of the flexible drill pipe was simulated, the contact force distribution of the flexible drill pipe during the motion cycle was analyzed, and the influence of the borehole curvature radius on the size of the flexible joints and the contact force was studied. The results show that the running form of the ball cage flexible drill pipe shows a “folded” shape compared with the initial form; the contact force of different flexible joints is in a state of fluctuation; the normal contact force is much larger than the tangential contact force; the matching relationship between the borehole curvature and the length and radius of the flexible joints is derived, which provides criteria for the design of the flexible joints to ensure the reliability of the flexible drill pipe in large curvature borehole; the borehole curvature has an important influence on the collision contact force and load transfer efficiency of flexible drill pipe.