  The fast beam-ion instability (FII) is caused by the interaction of an
electron bunch train with the residual gas ions. The ion oscillations in the
potential well of the electron beam have an inherent frequency spread due to
the nonlinear profile of the potential. However, this frequency spread and
associated with it Landau damping typically is not strong enough to suppress
the instability. In this work, we develop a model of FII which takes into
account the frequency spread in the electron beam due to the beam-beam
interaction in an electron-ion collider. We show that with a large enough
beam-beam parameter the fast ion instability can be suppressed. We estimate the
strength of this effect for the parameters of the eRHIC electron-ion collider.
