We study static solutions of the Tolman--Oppenheimer--Volkoff equations for spherically symmetric objects (stars) living in a space filled with the Chaplygin gas. Two cases are considered. In the normal case all solutions (excluding the de Sitter one) realize a three-dimensional spheroidal geometry because the radial coordinate achieves a maximal value (the "equator"). After crossing the equator, three scenarios are possible: a closed spheroid having a Schwarzschild-type singularity with infinite blue-shift at the "south pole", a regular spheroid, and a truncated spheroid having a scalar curvature singularity at a finite value of the radial coordinate. The second case arises when the modulus of the pressure exceeds the energy density (the phantom Chaplygin gas). There is no more equator and all solutions have the geometry of a truncated spheroid with the same type of singularity. We consider also static spherically symmetric configurations existing in a universe filled with the phantom Chaplygin gas only. In this case two classes of solutions exist: truncated spheroids and solutions of the wormhole type with a throat. However, the latter are not asymptotically flat and possess curvature singularities at finite values of the radial coordinate. Thus, they may not be used as models of observable compact astrophysical objects.