We consider a 3D volume preserving system inside the unit sphere. The system is a small perturbation of an integrable one. Almost all phase trajectories of the integrable system are closed curves. Under an arbitrarily small non-zero perturbation all the interior of the sphere, up to a residue of a small measure, is apparently a domain of chaotic motion. The phenomenon is described as a result of jumps in an adiabatic invariant of the system occurring when a phase trajectory crosses the 2D separatrix of the unperturbed (integrable) system. The dynamical properties can be understood as a consequence of splitting of invariant manifolds under the perturbation. A 2D return map generated by the system possesses strong stretching properties. This makes the dynamics of the system close to hyperbolic. Numerical investigation of the statistical properties of the system demonstrates a good agreement with theoretical predictions made under the assumption that the system has a big ergodic component whose measure tends to the measure of the sphere's interior as the perturbation tends to zero. However, the system is not ergodic, and stable periodic solutions are found, surrounded by stability islands of measure exponentially small with the perturbation.