  Using more than 100 galaxies in the MUSE Ultra Deep Field with spectroscopy
from the Hubble Space Telescope's Wide Field Camera 3 and the Very Large
Telescope's Multi Unit Spectroscopic Explorer, we extend the gas-phase
mass-metallicity relation (MZR) at $z\approx\,$1$\,$-$\,$2 down to stellar
masses of M$_{\star}$ $\approx$ 10$^{7.5}$ M$_{\odot}$. The sample reaches six
times lower in stellar mass and star formation rate (SFR) than previous HST
studies at these redshifts, and we find that galaxy metallicities decrease to
log(O/H) + 12 $\approx$ 7.8 $\pm$ 0.1 (15% solar) at
log(M$_{\star}$/M$_{\odot}$) $\approx$ 7.5, without evidence of a turnover in
the shape of the MZR at low masses. We validate our strong-line metallicities
using the direct method for sources with [O III] $\lambda$4363 and [O III]
$\lambda$1666 detections, and find excellent agreement between the techniques.
The [O III] $\lambda$1666-based metallicities double existing measurements with
S/N $\geq$ 5 for unlensed sources at $z~>$ 1, validating the strong-line
calibrations up to $z \sim$2.5. We confirm that the MZR resides $\sim$0.3 dex
lower in metallicity than local galaxies and is consistent with the fundamental
metallicity relation (FMR) if the low mass slope varies with SFR. At lower
redshifts ($z\sim$0.5) our sample reaches $\sim$0.5 dex lower in SFR than
current calibrations and we find enhanced metallicities that are consistent
with extrapolating the MZR to lower SFRs. Finally, we detect only a $\sim$0.1
dex difference in the metallicities of galaxies in groups versus isolated
environments. These results are based on robust calibrations and reach the
lowest masses and SFRs that are accessible with HST, providing a critical
foundation for studies with the Webb and Roman Space Telescopes.
