Pile installation has a great impact on the subsequent mechanical pile response. It is not, however, routinely incorporated in the numerical analyses of deep foundations in sand. Some of the difficulties associated with the simulation of the installation process are related to the fact that large deformations and distortions will eventually appear. The finite element method is not well suited to solve problems of this nature. Hence, an alternative procedure is tested herein, by using the material point method to simulate the installation of statically jacked or pushed-in type piles, which has successfully demonstrated its capacity to deal with this simulation. Two constitutive models were also tested, i.e., the modified Cam clay (MCC) and the subloading Cam clay (SubCam), allowing a clear perception of the great advantage to consider the soil with the SubCam model. The simulations have indeed reproduced some of the important features of the pile installation process, such as the radial stress acting around the pile’s shaft or the shaft’s lateral capacity, among other issues. The numerical results were additionally compared with known (semi-empirical) methods to derive the lateral capacity of the shaft, with a good and practical outcome.