We discuss the interaction of a quantum impurity with a one-dimensional degenerate Bose gas forming a Bose polaron. In three spatial dimension, the quasiparticle is typically well described by the extended Fröhlich model, in full analogy with the solid-state counterpart. This description, which assumes an undepleted condensate, fails, however, in 1D, where the backaction of the impurity on the condensate leads to a self-bound mean-field polaron for arbitrarily weak impurity-boson interactions. We present a model that takes into account this backaction and describes the impurity-condensate interaction as coupling to phononlike excitations of a deformed condensate. A comparison of polaron energies and masses to diffusion quantum Monte Carlo simulations shows very good agreement already on the level of analytical mean-field solutions and is further improved when taking into account quantum fluctuations.