In this study, we investigate the problem of tracking objects with unknown shapes using three-dimensional (3D) point cloud data. We propose a Gaussian process-based model to jointly estimate object kinematics, including position, orientation and velocities, together with the shape of the object for online and offline applications. We describe the unknown shape by a radial function in 3D, and induce a correlation structure via a Gaussian process. Furthermore, we propose an efficient algorithm to reduce the computational complexity of working with 3D data. This is accomplished by casting the tracking problem into projection planes which are attached to the object's local frame. The resulting algorithms can process 3D point cloud data and accomplish tracking of a dynamic object. Furthermore, they provide analytical expressions for the representation of the object shape in 3D, together with confidence intervals. The confidence intervals, which quantify the uncertainty in the shape estimate, can later be used for solving the gating and association problems inherent in object tracking. The performance of the methods is demonstrated both on simulated and real data. The results are compared with an existing random matrix model, which is commonly used for extended object tracking in the literature.