Abstract <p>For the first time, the paper considers the dynamic contact problem of nonstationary impact in the contact zone of a flexible die on the surface of a deformable multilayer medium. The contact problem is considered in a two-dimensional formulation. The die is assumed to be semi-infinite and acts over a semi-infinite time interval. Traditionally, in such dynamic contact problems, expansion into a double series or Fourier integral of a time-varying function describing the behavior of the die base is used. After that, a contact problem with excluded exponential functions containing a time parameter is considered. However, it is known that such Fourier series do not effectively describe processes accompanied by violation of the smoothness of functions, in particular, if they include singularities. The problem is solved sequentially: first in the geometric parameters and then the time dependence is investigated. This did not allow the subtle features of the dependence of the solutions on the time parameter to be revealed. In the current work, this shortcoming is eliminated. The case of a two-dimensional problem in which geometric and time parameters are equally included is considered. The contact problem is reduced to the two-dimensional Wiener–Hopf integral equation, the solution method of which has been developed recently. The resulting solution, depending on the geometric and time parameters, allows identification of the previously undescribed effect of a temporary surge at the initial time instance of the contact stresses under the die. The result allows, by adjusting the time of impact of the die on the medium, selection of the optimal modes of emerging contact stresses. The method enables generalization to contact problems of higher dimensions.</p>