The main purpose of our paper is to construct a viable Kaluza-Klein model satisfying the observational constraints. To this end, we investigate a six-dimensional model with spherical compactification of the internal space. Background matter is considered in the form of a perfect fluid with nonlinear equations of state both in the external/our and internal spaces, and the model is set to include an additional bare cosmological constant Lambda(6). In the weak-field approximation, the background is perturbed by a pressureless gravitating mass that is a static pointlike particle. The nonlinearity of the equations of state of the perfect fluid makes it possible to solve simultaneously a number of problems. The requirement that the post-Newtonian parameter gamma be equal to 1 in this configuration, first, ensures compatibility with the gravitational tests in the Solar system (deflection of light and time delay of radar echoes) at the same level of accuracy as General Relativity. Second, it translates into the absence of internal space variations, so that the gravitational potential exactly coincides with the Newtonian one, securing the absence of a fifth force. Third, the gravitating mass remains pressureless in the external space, as in the standard approach to nonrelativistic astrophysical objects and, meanwhile, acquires an effective tension in the internal space.