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The Ba\~{n}ados-Silk-West (BSW) effect consists in the possibility to obtainarbitrarily large energy $E_{c.m.}$ in the centre of mass frame of twocolliding particles near the black hole horizon. One of the common beliefs wasthat the action of force on these particles (say, due to gravitationalradiation) should necessarily restrict the growth of $E_{c.m.}$. We considerextremal horizons and develop a model-independent approach and analyze theconditions for the force to preserve or kill the effect, using the framesattached both to observers orbiting the black hole and to ones crossing thehorizon. We argue that the aforementioned expectations are not confirmed. Underrather general assumptions, the BSW effect survives. For equatorial motion itis only required that in the proper frame the radial component of the force befinite, while the azimuthal one tend to zero not too slowly. If the lattercondition is violated, we evaluate $E_{c.m.}$, which becomes indeed restrictedbut remains very large for small forces.

Ba\~{n}ados-Silk-West effect with nongeodesic particles: extremal horizons