We present new measurements of the very low-mass end of the galaxy stellar mass function (GSMF) at $zsim6-7$ computed from a rest-frame ultraviolet selected sample of dropout galaxies. These galaxies lie behind the six Hubble Frontier Fields clusters and are all gravitationally magnified. Using deep Spitzer/IRAC and Hubble Space Telescope imaging, we derive stellar masses by fitting galaxy spectral energy distributions and explore the impact of different model assumptions and parameter degeneracies on the resulting GSMF. Our sample probes stellar masses down to $M_{star}>10^{6},text{M}_{odot}$ and we find the $zsim6-7$ GSMF to be best parametrized by a modified Schechter function which allows for a turnover at very low masses. Using a Monte-Carlo Markov Chain analysis of the GSMF, including accurate treatment of lensing uncertainties, we obtain a relatively steep low-mass end slope $alphasimeq-1.96_{-0.08}^{+0.09}$ and a turnover at $log(M_T/text{M}_{odot})simeq7.10_{-0.56}^{+0.17}$ with a curvature of $betasimeq1.00_{-0.73}^{+0.87}$ for our minimum assumption model with constant star-formation history (SFH) and low dust attenuation, $A_Vleq0.2$. We find that the $zsim6-7$ GSMF, in particular its very low-mass end, is significantly affected by the assumed functional form of the star formation history and the degeneracy between stellar mass and dust attenuation. For example, the low-mass end slope ranges from $alphasimeq-1.82_{-0.07}^{+0.08}$ for an exponentially rising SFH to $alphasimeq-2.34_{-0.10}^{+0.11}$ when allowing $A_V$ of up to 3.25. Future observations at longer wavelengths and higher angular resolution with the James Webb Space Telescope are required to break these degeneracies and to robustly constrain the stellar mass of galaxies on the extreme low-mass end of the GSMF.
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