The detectors of the direct dark matter search experiment EDELWEISS consist of high-purity germanium crystals operated at cryogenic temperatures ( $\mathrm {{<}20\,mK}$ ) and low electric fields ( $\mathrm {{<}1\,V/cm}$ ). The surface discrimination is based on the simultaneous measurement of the charge amplitudes on different sets of electrodes. As the rise time of a charge signal strongly depends on the location of an interaction in the crystal, a time-resolved measurement can also be used to identify surface interactions. This contribution presents the results of a study of the discrimination power of the rise time parameter from a hot carrier transport simulation in combination with time-resolved measurements using an EDELWEISS-type detector in a test cryostat at ground level. We show the setup for the time-resolved ionization signal read-out in the EDELWEISS-III experiment and first results from data taking in the underground laboratory of Modane.