We present an analysis of the properties of the lowest H alpha-luminosity galaxies (L-H alpha <4 x 1032 W; SFR <0.02 M-circle dot yr-1, with SFR denoting the star formation rate) in the Galaxy And Mass Assembly survey. These galaxies make up the rise above a Schechter function in the number density of systems seen at the faint end of the H alpha luminosity function. Above our flux limit, we find that these galaxies are principally composed of intrinsically low stellar mass systems (median stellar mass = 2.5 x 108 M-circle dot) with only 5/90 having stellar masses M > 1010 M-circle dot. The low-SFR systems are found to exist predominantly in the lowest-density environments (median density similar to 0.02 galaxy Mpc-2) with none in environments more dense than similar to 1.5 galaxy Mpc-2. Their current specific SFRs (SSFRs; -8.5 <log [SSFR (yr -1)] <-12) are consistent with their having had a variety of star formation histories. The low-density environments of these galaxies demonstrate that such low-mass, star-forming systems can only remain as low mass and form stars if they reside sufficiently far from other galaxies to avoid being accreted, dispersed through tidal effects or having their gas reservoirs rendered ineffective through external processes.