In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem. For the displacement, we use the pushforward of a NURBS space of degree p and for the Lagrange multiplier, the pushforward of a B-Spline space of degree p−2. These choices of space ensure to prove an inf−sup condition and so on, the stability of the method. We distinguish between contact and non-contact sets to avoid of classical geometrical hypothesis of the contact set. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two- and three-dimensions and in small and large deformation demonstrate the accuracy of the proposed method.