This work presents a so-called mixed interpolation method (MIP) in the context of contact formulations. Its implementation is a simple extension from the standard displacement-based (penalty-based, Gauss-Point-To-Segment) contact formulation, yet it is shown to enhance the robustness of contact simulations quite significantly. The enhanced robustness allows for using larger load step size (for efficiency), and a larger penalty parameter (for accuracy in the contact constraint enforcement). The basic idea of the proposed MIP contact method is relaxing the contact constitution at quadrature points. At first, the derivation of the formulation considers the contact pressure as an additional unknown apart from the displacement field and the perturbed Lagrange multiplier potential is used to enforce the contact condition. However, the unknown contact pressure is then eliminated directly at quadrature points, which results in the residual vector identical to the standard displacement-based contact formulation. Their difference lies in the computation of the tangent stiffness matrix: the MIP tangents are based on an extrapolation of the contact pressure iteratively over Newton iterations. In order to avoid discontinuities of the normal contact gap, we employ a smooth isogeometric discretization for the contact surface. We consider frictionless contact in this work. The robustness and accuracy of the proposed formulation will be demonstrated by several challenging numerical examples.