The description of the three-nucleon system in terms of nucleon and Δ degrees of freedom is extended to allow for explicit pion production (absorption) from single dynamic Δ deexcitation (excitation) processes. This mechanism yields an energy dependent effective three-body Hamiltonian. The Faddeev equations for the trinucleon bound state are solved with a force model that has already been tested in the two-nucleon system above pion-production threshold. The binding energy and other bound-state properties are calculated. The contribution to the effective three-nucleon force arising from the pionic degrees of freedom is evaluated. The validity of previous coupled-channel calculations with explicit but stable Δ isobar components in the wave function is studied.