We study how an external bias field influences the Brownian particle survival in a medium with traps. The emphasis is on the many-body aspect of the problem. A general path integral approach is used to present the particle survival probability P(t) in the form providing a uniform description of the process over the whole range of time and for any value of the field strength. It is shown that at low fields the many-body effects determine the long-time behavior of P(t). At high fields, they manifest themselves only as a small correction to the rate constant predicted by the single-body theory. Particular attention is given to the one-dimensional case where an exact solution can be obtained. A major observation is that the difference between the exact and the mean-field expressions for P(t) (considered as a measure of the magnitude of the many-body effects) behaves nonmonotonically with the field. This suggests that an optimal choice of the field may facilitate an experimental observation of the many-body effects in the trapping kinetics.