It is well known, that the dynamics of small particles moving in a viscous fluid is strongly influenced by the long-range hydrodynamical interaction between them. Motion at high viscosity is usually treated by means of the Stokes equations, which are linear and instantaneous. Nevertheless, the hydrodynamical interaction mediated by the liquid is nonlinear; therefore the dynamics of more than two particles can be rather complex. Here we present a high resolution numerical analysis of the classical three-particle Stokeslet problem in a vertical plane. We show that a chaotic saddle in the phase space is responsible for the extreme sensitivity to initial configurations, which has been mentioned several times in the literature without an explanation. A detailed analysis of the transiently chaotic dynamics and the underlying fractal patterns is given.