Using Cl- as a test case, Wigner time delay in the photodetachment process has been investigated theoretically, along with the photoionization of the isoelectronic Ar atom, for the outer 3p shell using the relativistic-random-phase approximation (RRPA). Time delay was probed in these systems from threshold to 80 eV, to investigate threshold effects, the centrifugal barrier shape resonance, and the Cooper minimum region. This study focuses on Cl- because, for negative ions, the phase of the photoemission process is not dominated by the Coulomb phase as it is in photoionization. The results show significant differences, both qualitative and quantitative, between the time delays for Cl- and Ar photoemission at low photoelectron energy, but they are rather similar in the Cooper minimum region, where the Coulomb phase is small. In particular, the Wigner time delay in Cl- exhibits a dramatic energy dependence just above threshold, and a rapidly increasing time delay in the vicinity of the shape resonance. A strong angular dependence of time delay has also been found near the threshold region for the Cl- case, and is absent in the case of the photoionization of Ar. The origin of these phenomenologies is explained and a prospectus for future work is presented. © 2019 American Physical Society.