We consider the Cauchy problem of cooperative reaction-diffusion systems with nonnegative initial data. Here we discuss the blow-up of a solution that occurs only at space infinity, We give sufficient conditions for such phenomena, and study an asymptotic behaviour at space infinity of the solutions at the blow-up time. In general, relatively little is proved on the locations of blow-up point for semilinear parabolic systems. However, our results can be applied to a large class of nonlinearity for some class of initial value. The reason of this is that, when the blow-up occurs only at space infinity, the effect of reaction is much stronger than that of diffusion, and the behaviour at space infinity is well approximated by the flat solution.