The analytical properties of the pion form factor in the complex plane of momentum transfers are discussed in the Poincaré-invariant quark model. It is shown, that these analytical properties are similar to ones following from basic principles of quantum field theory. The strong dependence of pion form factor on the type of the pion wave functions is obtained in the time-like momentum transfers region. The conditions of resonance behavior of the form factor in this region are formulated in our model. The simple examples of wave functions giving the resonance behavior of the form factor are constructed.