Non-rigid shape deformations pose significant challenges, and most existing methods struggle to handle partial deformations effectively. We present Partial Non-rigid Deformations and interpolations of the human body Surfaces (PaNDAS), a new method to learn local and global deformations of 3D surface meshes by building on recent deep models. Unlike previous approaches, our method enables restricting deformations to specific parts of the shape in a versatile way and allows for mixing and combining various poses from the database, all while not requiring any optimization at inference time. We demonstrate that the proposed framework can be used to generate new shapes, interpolate between parts of shapes, and perform other shape manipulation tasks with state-of-the-art accuracy and greater locality across various types of human surface data. Code and data will be made available soon.