A positron timing counter (TC) with 30-40 ps time resolution for 50 MeV/c positrons required by the MEG II experiment has been developed. We employed a highly segmented design with 512 scintillator plates (120 × 40 × 5 mm³ and 120 × 50 × 5 mm³) read out by 6-SiPM-arrays at the both ends. Pile up is reduced by the segmented design and multi-counter measurement improves the overall timing resolution. All the single counters were assembled and their resolutions were measured to be below 100 ps in a pre-test with a ⁹⁰Sr source. The construction and installation were completed in 2017. An engineering run was performed at the end of 2017 at the πE5 muon beam line in PSI, which supplies the most intense muon beam (7 × 10⁷ stops/s on a target) in the world. We successfully operated the full system of TC in the MEG II environment and achieved a time resolution below 40 ps with more than 6 counter hits. The overall resolution for the signal positrons is estimated to be 38.5 ps by weighting the obtained resolutions with the number of hits distribution of the signal positrons obtained by a Monte Carlo simulation. The time resolution of the TC is improved by a factor of 2 with respect to the MEG experiment.