Experiments were conducted to clarify role of the outermost edge flame on low-strain-rate flame extinction in buoyancy-suppressed non-premixed methane flames diluted with He and $N_2$. The use of He curtain flow produced a microgravity level of $10^{-2}-10^{-3}g$ in $N_2$- and He-diluted non-premixed counterflow flame experiments. The critical He and $N_2$ mole fractions at extinction with a global strain rate were examined at various burner diameters (10, 20, and 25 mm). The results showed that the extinction curves differed appreciably with burner diameter. Before the turning point along the extinction curve, low-strain-rate flames were extinguished via shrinkage of the outermost edge flame with and without self-excitation. High-strain-rate flames were extinguished via a flame hole while the outermost edge flame was stationary. These characteristics could be identified by the behavior of the outermost edge flame. The results also showed that the outermost edge flame was not influenced by radiative heat loss but by convective heat addition and conductive heat losses to the ambient He curtain flow. The numerical results were discussed in detail. The self-excitation before the extinction of a low-strain-rate flame was well described by a dependency of the Strouhal number on global strain rate and normalized nozzle exit velocity.