The paper proposes a method for determining the mechanical properties of a highly elastic material by indenting a thin round plate. The novelty of the proposed method is to take into account the influence of friction in the contact area of the indenter and the specimen. The mathematical model is based on the theory of nonlinear-elastic membranes and the Coulomb friction model. The membrane is made of an isotropic incompressible material. The indentation process is considered quasi-static and the membrane deformation is axisymmetric. The problem is reduced to a boundary value problem for two systems of ordinary differential equations with a parameter (unknown contact boundary of the indenter and membrane). The boundary value problem is solved by the shooting method. Numerical analyses are carried out for the neo-Hookean model of the material. Based on the numerical results, the “indentation force — indenter displacement” curve is approximated by a polynomial expression. The determining of the material constant is based on minimising the difference between the experimental “force — displacement” curve and the approximating expression. The method is validated on experimental data of indentation of a thin rubber band under different contact conditions (without lubrication and with lubrication). For this purpose, experiments were carried out: on determination of the friction coefficient, on indentation, uniaxial and uniform biaxial stretching. The value of the restored material constant determined from the indentation experiment is close to the results of classical methods. In the case when friction is not taken into account during modeling, the value of the material constant will be significantly overestimated.