An experiment has been performed to populate several extremely neutron-deficient nuclei around the mass-140 region of the nuclear chart, using a beam of 54 Fe on a 92 Mo target at an energy of 315 MeV. Analysis of these data using recoil-isomer tagging has established that the yrast πh 11/2 ⊗ νh 11/2 , J π = (8 + ), bandhead state in 136 Pm is isomeric with a half-life of 1.5(1) µs. This isomeric state decays via a 43-keV, probable-E1 transition to a J π = (7 − ) state. Consideration of the theoretical Nilsson orbitals near the Fermi surface suggests that the J π = (8 + ) state has a νh 11/2 [505] 11 2 − ⊗ πh 11/2 [532] 5 2 − configuration, which decays to the J π = (7 − ) state with a νh 11/2 [505] 11 2 − ⊗ πd 5/2 [411] 3 2 + configuration. Differences in the shape-driving effects for these two configurations is reasoned to be responsible for the long half-life of the J π = (8 + ) isomeric state. The non-observation of other γ rays in prompt or delayed coincidence with the 43-keV transition suggests that this transition may feed another, longer lived isomeric state with a half-life of the order of milliseconds or greater. However, the present experiment was not sensitive to the decay of this new J π = (7 − ) state by internal conversion or even β decay.