We use spin-torque resonance to probe simultaneously and separately the dynamics of a magnetic domain wall and of magnetic domains in a nanostripe magnetic tunnel junction. Thanks to the large associated resistance variations we are able to analyze quantitatively the resonant properties of these single nanoscale magnetic objects. In particular, we find that the magnetic damping of both domains and domain walls is doubled compared to the damping value of their host magnetic layer. We estimate the contributions to damping arising from dipolar couplings between the different layers in the junction and from the intralayer spin pumping effect. We find that they cannot explain the large damping enhancement that we observe. We conclude that the measured increased damping is intrinsic to large amplitudes excitations of spatially localized modes or solitons such as vibrating or propagating domain walls