We propose a model based on extreme value statistics (EVS) and combine it with different models for single-asperity contact, including adhesive and elasto-plastic contacts, to derive a relation between the applied load and the friction force on a rough interface. We determine that, when the summit distribution is Gumbel and the contact model is Hertzian, we obtain the closest conformity with Amonton’s law. The range over which Gumbel distribution mimics Amonton’s law is wider than that of the Greenwood–Williamson (GW) model. However, exact conformity with Amonton’s law is not observed for any of the well-known EVS distributions. Plastic deformations in the contact area reduce the relative change in pressure slightly with Gumbel distribution. Interestingly, when elasto-plastic contact is assumed for the asperities, together with Gumbel distribution for summits, the best conformity with Amonton’s law is achieved. Other extreme value statistics are also studied, and the results are presented. We combine Gumbel distribution with the GW–McCool model, which is an improved version of the GW model, and the new model considers a bandwidth for wavelengths α. Comparisons of this model with the original GW–McCool model and other simplified versions of the Bush–Gibson–Thomas theory reveal that Gumbel distribution has a better conformity with Amonton’s law for all values of α. When the adhesive contact model is used, the main observation is that there is some friction for zero or even negative applied load. Asperities with a height even less than the separation between the two surfaces are in contact. For a small value of the adhesion parameter, a better conformity with Amonton’s law is observed. The relative pressure increases for stronger adhesion, which indicates that adhesion-controlled friction is dominated by load-controlled friction. We also observe that adhesion increases on a surface with a lower value of roughness.