Spike propagation in chain networks is usually studied in the synfire regime, in which successive groups of neurons are synaptically activated sequentially through the unidirectional excitatory connections. Here we study the dynamics of chain networks with dominant global feedback inhibition that prevents the synfire activity. Neural activity is driven by suprathreshold external inputs. We analytically and numerically demonstrate that spike propagation along the chain is a unique dynamical attractor in a wide parameter regime. The strong inhibition permits a robust winner-take-all propagation in the case of multiple chains competing via the inhibition.