  We consider a generalized discrete $\phi^4$ model and demonstrate that it can
support exact moving kink solutions in the form of tanh with an arbitrarily
large velocity. The constructed exact moving solutions are dependent on the
specific value of the propagation velocity. We demonstrate that in this class
of models, given a specific velocity, the problem of finding the exact moving
solution is integrable. Namely, this problem originally expressed as a
three-point map can be reduced to a two-point map, from which the exact moving
solutions can be derived iteratively. It was also found that these high-speed
kinks can be stable and robust against perturbations introduced in the initial
conditions.
