Force-displacement curves have been obtained with a commercial atomic force microscope (AFM) at different temperatures and probe rates on a thick film of poly(n-butyl methacrylate) (PnBMA). The analysis of the force-displacement curves has been focused on the contact portion of the curves, giving information about the stiffness of the sample and its Young's modulus. A novel model of sample deformations that extends the basic equations of the elastic continuum contact theories to the plastic deformations is presented. This model gives several insights into the processes of deformation of soft samples and permits to calculate not only the parameters of the Williams-Landel-Ferry equation but also the Young's modulus and the yielding force of the polymer as a function of temperature and/or probe rate. These quantities have been measured in a wide range of temperatures (70 K) and probe rates (6 decades) for the first time with the AFM, and the results are in very good agreement with measurements performed with customary techniques, such as broadband spectroscopy and dynamic mechanical analysis.