We examined the spatial dynamics of in vitro clot growth in human blood and plasma and found that initially, a clot grows at a constant speed, then abruptly stops and becomes surrounded by an 'inhibition zone' in which coagulation is strongly suppressed. We also observed the formation of 'stratified structures' (target patterns) in which solid layers alternated with liquid plasma. These and other spatial regimes of clotting are explained in terms of two interacting concentration waves propagating without attenuation. The experimental results are consistent with a hypothesis that blood is a bi-excitable medium, a new type of excitable medium.