  We present new measurements of the very low-mass end of the galaxy stellar
mass function (GSMF) at $z\sim6-7$ 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 $z\sim6-7$ 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
$\alpha\simeq-1.96_{-0.08}^{+0.09}$ and a turnover at
$\log(M_T/\text{M}_{\odot})\simeq7.10_{-0.56}^{+0.17}$ with a curvature of
$\beta\simeq1.00_{-0.73}^{+0.87}$ for our minimum assumption model with
constant star-formation history (SFH) and low dust attenuation, $A_V\leq0.2$.
We find that the $z\sim6-7$ 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 $\alpha\simeq-1.82_{-0.07}^{+0.08}$
for an exponentially rising SFH to $\alpha\simeq-2.34_{-0.10}^{+0.11}$ 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.
