Nonlinear deformations of a clamped-edge nano-film are considered on the basis of an extension of von Kármán’s theory of elastic plate, taken into account the Gurtin–Murdoch surface elasticity and Kirchhoff’s hypothesis. Unlike most of the previous related theories, surface tension, usually omitted in the conjugation condition of Young–Laplace law in the transverse direction, is incorporated in the two-dimensional motion and constitutive relations for membrane forces together with quadratic terms equal to the von Kármán-type strains. The influence of the linear and nonlinear terms of surface tension is illustrated in the cases of nonlinear bending, post-critical compressive buckling, and free transverse vibration of a striplike nano-film with clamped edges.