The phenomenon of herding is a very general feature of the collective behavior of many species in panic conditions, including humans. It has been predicted theoretically that panic-induced herding in individuals confined to a room can produce a nonsymmetrical use of two identical exit doors. Here we demonstrate the existence of that phenomenon in experiments, using ants as a model of pedestrians. We show that ants confined to a cell with two symmetrically located exits use both exits in approximately equal proportions to abandon it in normal conditions but prefer one of the exits if panic is created by adding a repellent fluid. In addition, we are able to reproduce the observed escape dynamics in detail using a modification of a previous theoretical model that includes herding associated with a panic parameter as a central ingredient. Our experimental results, combined with theoretical models, suggest that some features of the collective behavior of humans and ants can be quite similar when escaping under panic.