Beta-delayed-neutron (βn) spectroscopy was performed using the Beta-decay Paul Trap and an array of radiation detectors. The βn branching ratios and energy spectra for 135,136 Sb and 140 I were obtained by measuring the time of flight of recoil ions emerging from the trapped ion cloud. These nuclei are located at the edge of an isotopic region identified as having βn branching ratios that impact the r -process abundance pattern around the A∼130 peak. For 135,136 Sb and 140 I, βn branching ratios of 14.6(11)%, 17.6(28)%, and 7.6(28)% were determined, respectively. The βn energy spectra obtained for 135 Sb and 140 I are compared with results from direct neutron measurements, and the βn energy spectrum for 136 Sb has been measured for the first time.