Angle-resolved photoemission (ARPES) on the quasi-one-dimensional conductor (TaSe4)(2)I shows a hidden Fermi-surface crossing in its metallic state and the opening of a Peierls gap at low temperatures, The underlying quasiparticles have vanishing spectral weight and extremely short coherence lengths. They are interpreted as polarons in the strong-coupling adiabatic Lin-Lit, and almost all their ARPES weight is incoherent. These observations suggest a scenario where the long-standing contradictions between ARPES and other experiments on Peierls materials could be resolved.