Context. The Small Magellanic Cloud (SMC) hosts many known high-mass X-ray binaries (HMXBs), and all but one (SMC X-1) have a Be companion star. Through the calibration and verification phase of eROSITA on board the Spektrum-Roentgen-Gamma (SRG) spacecraft, the Be/X-ray binary XMMU J010429.4-723136 was in the field of view during observations of the supernova remnant, 1E0102.2-7219, used as a calibration standard. Aims. We report timing and spectral analyses of XMMU J010429.4-723136 based on three eROSITA observations of the field, two of which were performed on 2019 November 7-9, with the third on 2020 June 18-19. We also reanalyse the OGLE-IV light curve for that source in order to determine the orbital period. Methods. We performed a Lomb-Scargle periodogram analysis to search for pulsations (from the X-ray data) and for the orbital period (from the OGLE data). X-ray spectral parameters and fluxes were retrieved from the best-fit model. Results. We detect, for the first time, the pulsations of XMMU J010429.4-723136 at a period of -164 s, and therefore designate the source as SXP 164. From the spectral fitting, we derive a source flux of ~1 × 10 −12 erg s −1 cm −2 for all three observations, corresponding to a luminosity of ~4 × 10 35 erg s −1 at the distance of the SMC. Furthermore, reanalysing the OGLE light curve, including the latest observations, we find a significant periodic signal that we believe is likely be the orbital period; at 22.3 days, this is shorter than the previously reported values. The Swift/XRT light curve, extracted from two long monitorings of the field and folded at the same period, suggests that a modulation is also present in the X-ray data.