  The Ba\~{n}ados-Silk-West (BSW) effect consists in the possibility to obtain
arbitrarily large energy $E_{c.m.}$ in the centre of mass frame of two
colliding particles near the black hole horizon. One of the common beliefs was
that the action of force on these particles (say, due to gravitational
radiation) should necessarily restrict the growth of $E_{c.m.}$. We consider
extremal horizons and develop a model-independent approach and analyze the
conditions for the force to preserve or kill the effect, using the frames
attached both to observers orbiting the black hole and to ones crossing the
horizon. We argue that the aforementioned expectations are not confirmed. Under
rather general assumptions, the BSW effect survives. For equatorial motion it
is only required that in the proper frame the radial component of the force be
finite, while the azimuthal one tend to zero not too slowly. If the latter
condition is violated, we evaluate $E_{c.m.}$, which becomes indeed restricted
but remains very large for small forces.
