The Virtual Element Method (VEM) is a numerical technique that can be viewed as an extended form of the Finite Element Method. In the prior studies of the VEM, a stabilization term is required in order to ensure that the global stiffness matrix has the correct rank. In this work, we present a stabilization-free Virtual Element Method for compressible hyper-elastic materials in 3D. The main idea of the stabilization-free approach is to use an enhanced approximation space to compute a higher-order polynomial $L_2$ projection of the gradient. The formulation of the stabilization-free VEM for hyper-elastic materials is given in detail. The arc-length method is selected for unstable structures. Some numerical examples are given to compare the accuracy of the stabilization-free VEM with the conventional VEM.