We study how star formation is regulated in low-mass field dwarf galaxies (⁠|$10^5 \le M_{\star } \le 10^6 \, \mbox{M}_\mathrm{\odot }$|⁠), using cosmological high-resolution (⁠|$3 \, \mathrm{pc}$|⁠) hydrodynamical simulations. Cosmic reionization quenches star formation in all our simulated dwarfs, but three galaxies with final dynamical masses of |$3 \times 10^{9} \, \mbox{M}_\mathrm{\odot }$| are subsequently able to replenish their interstellar medium by slowly accreting gas. Two of these galaxies reignite and sustain star formation until the present day at an average rate of |$10^{-5} \, \mbox{M}_\mathrm{\odot } \, \text{yr}^{-1}$|⁠, highly reminiscent of observed low-mass star-forming dwarf irregulars such as Leo T. The resumption of star formation is delayed by several billion years due to residual feedback from stellar winds and Type Ia supernovae; even at z = 0, the third galaxy remains in a temporary equilibrium with a large gas content but without any ongoing star formation. Using the ‘genetic modification’ approach, we create an alternative mass growth history for this gas-rich quiescent dwarf and show how a small |$(0.2\, \mathrm{dex})$| increase in dynamical mass can overcome residual stellar feedback, reigniting star formation. The interaction between feedback and mass build-up produces a diversity in the stellar ages and gas content of low-mass dwarfs, which will be probed by combining next-generation H i and imaging surveys.