Computational contact mechanics is a challenging yet well-studied problem within the framework of classical finite element methods (FEM). Isogeometric Analysis (IGA), which inherited many contact solution schemes from the classical FEM framework, leverages its higher smoothness and geometric exactness to improve contact resolution accuracy further. However, solving elastodynamics problems involving complex mutual and self-contact among numerous deformable bodies remains challenging, even for IGA. We propose the Immersed Incremental Potential Contact (IIPC) method to address general contact problems in IGA. The work is motivated by the challenges of resolving contact problems using the immersed boundary method. The original and high-order formulations of the Incremental Potential Contact (IPC) algorithm are detailed in [1, 2]. The core concept involves constructing a set of linear triangular meshes by upsampling the NURBS surfaces of the deformable bodies. The IPC algorithm robustly solves the contact sub-problem between deformable bodies by using this triangular mesh set as collision proxies. Meanwhile, the nonlinear elasticity sub-problem is addressed within the total Lagrangian framework and solved using the immersed isogeometric method. We adopted the formulation introduced by Meßmer et al. [3] for the immersed analysis. Combining the immersed isogeometric analysis and the incremental potential contact algorithm exhibits remarkable capabilities in resolving general contact problems, significantly improving accuracy, versatility, and robustness. Furthermore, decoupling the mesh used for modeling elasticity from the mesh used for contact introduces significant flexibility: the collision proxy mesh can be defined with a resolution much higher or lower than that of the volumetric elasticity mesh. This decoupling allows explicit control over the accuracy and efficiency of the simulation by adjusting the sampling of the collision mesh. We will demonstrate test cases where the proposed method ensures global convergence and non-penetration for solids with arbitrary geometries and topologies. Additionally, we have applied the technique to scenarios involving extreme deformations and dramatic contact interactions, which are rarely addressed in existing IGA literature.