Attosecond time delays in molecular photoionization at low energies are dominated by the contribution of the Coulomb phase, which diverges as the energy of the photoelectron decreases towards zero. With calculations on {core-electron ejection from} CN${}^{-}$ and C${}_2^{-}$ we demonstrate that in photodetachment of molecular anions the absence of the long-range Coulomb interaction in the final state can reveal the delays of narrow low-energy shape resonances that do not generally occur in photoionization. Delays of several femtoseconds can be associated with such resonances. The angular dependence of photodetachment time delays in the body frame is highly sensitive to short-range anisotropies in the electron-molecule interaction {as well as to the initial state}.