  The experimental violation of Bell inequalities using spacelike separated
measurements precludes the explanation of quantum correlations through causal
influences propagating at subluminal speed. Yet, any such experimental
violation could always be explained in principle through models based on hidden
influences propagating at a finite speed v>c, provided v is large enough. Here,
we show that for any finite speed v with c<v<infinity, such models predict
correlations that can be exploited for faster-than-light communication. This
superluminal communication does not require access to any hidden physical
quantities, but only the manipulation of measurement devices at the level of
our present-day description of quantum experiments. Hence, assuming the
impossibility of using nonlocal correlations for superluminal communication, we
exclude any possible explanation of quantum correlations in terms of influences
propagating at any finite speed. Our result uncovers a new aspect of the
complex relationship between multipartite quantum nonlocality and the
impossibility of signalling.
