The Metal Abundances across Cosmic Time ($\mathcal{MACT}$) Survey. III
  -- The relationship between stellar mass and star formation rate in extremely
  low-mass galaxies
  Extragalactic studies have demonstrated there is a moderately tight
($\approx$0.3 dex) relationship between galaxy stellar mass ($M_{\star}$) and
star formation rate (SFR) that holds for star-forming galaxies at $M_{\star}
\sim 3 \times 10^8$-10$^{11}~M_{\odot}$, i.e., the "star formation main
sequence." However, it has yet to be determined whether such a relationship
extends to even lower mass galaxies, particularly at intermediate or higher
redshifts. We present new results using observations for 714 narrowband
H$\alpha$-selected galaxies with stellar masses between $10^6$ and
$10^{10}~M_{\odot}$ (average of $10^{8.2}~M_{\odot}$) at $z \approx$ 0.07-0.5.
These galaxies have sensitive UV to near-infrared photometric measurements and
optical spectroscopy. The latter allows us to correct our H$\alpha$ SFRs for
dust attenuation using Balmer decrements. Our study reveals: (1) for low-SFR
galaxies, our H$\alpha$ SFRs systematically underpredict compared to FUV
measurements, consistent with other studies; (2) at a given stellar mass
($\approx $10$ ^{8}~M_{\odot}$), log(specific SFR) evolves as $ A \log(1+z) $
with $ A = 5.26 \pm 0.75 $, and on average, specific SFR increases with
decreasing stellar mass; (3) the SFR-$M_{\star}$ relation holds for galaxies
down to $\sim$10$^6~M_{\odot}$ ($\sim$1.5 dex below previous studies), and over
lookback times of up to 5 Gyr, follows a redshift-dependent relation of
$\log{({\rm SFR})} \propto \alpha \log(M_{\star}/M_{\odot}) + \beta z$ with
$\alpha = 0.60 \pm 0.01$ and $\beta = 1.86 \pm 0.07$; and (4) the observed
dispersion in the SFR-$M_{\star}$ relation at low stellar masses is
$\approx$0.3 dex. Accounting for survey selection effects using simulated
galaxies, we estimate the true dispersion is $\approx$0.5 dex.
