Phase spaces with nontrivial geometry appear in different approaches to quantum gravity and can also play a role in e.g. condensed matter physics. However, so far, such phase spaces have only been considered for particles or strings. We propose an extension of the usual field theories to the framework of fields with nonlinear phase space of field values, which generally means nontrivial topology or geometry. In order to examine this idea, we construct a prototype scalar field with the spherical phase space and then study its quantized version with the help of perturbative methods. As a result, we obtain a variety of predictions that are known from the quantum gravity research, including algebra deformations, generalization of the uncertainty relation and shifting of the vacuum energy.