The pressure dependence of the metastable-state lifetimes in antiprotonic 3He atoms p¯-3He+ was studied using the recently observed laser resonance transition (n,l)=(36,33)-->(35,32) in the υ=n-l-1=2 cascade. To this end antiprotons from the Low Energy Antiproton Ring (LEAR) at CERN were extracted in approximately 200-ns-long bunches with about 108 p¯ per bunch and stopped in a 3He gas target. Time spectra of delayed annihilation products were taken with the help of a Čerenkov counter with a gated photomultiplier. Under our experimental conditions, p=136-690 mbars and T=5.8 K, the lowest metastable level (36,33) was found to be much shorter lived than the corresponding state (37,34) in the υ=2 cascade of p¯-4He+. It was strongly quenched at pressures above 500 mbars, whereas the higher states in the cascade remained nearly unaffected. This enabled us to investigate the (37,34)-->(36,33) transition by resonant laser deexcitation. Its wavelength was found to be λ=524.155+/-0.004 nm, only 6 ppm lower than the theoretical value given by Korobov and Bakalov [Phys. Rev. Lett. 79, 3379 1997)] after including relativistic corrections.