  We present determinations of the gas-phase and stellar metallicities of a
sample of 65 star-forming galaxies at $z \simeq 3.5$ using rest-frame
far-ultraviolet (FUV) spectroscopy from the VANDELS survey in combination with
follow-up rest-frame optical spectroscopy from VLT/KMOS and Keck/MOSFIRE. We
infer gas-phase oxygen abundances ($Z_{\mathrm{g}}$; tracing O/H) via strong
optical nebular lines and stellar iron abundances ($Z_{\star}$; tracing Fe/H)
from full spectral fitting to the FUV continuum. Our sample spans the stellar
mass range $8.5 < \mathrm{log}(M_{\star}/\mathrm{M}_{\odot}) < 10.5$ and shows
clear evidence for both a stellar and gas-phase mass-metallicity relation
(MZR). We find that our O and Fe abundance estimates both exhibit a similar
mass-dependence, such that $\mathrm{Fe/H}\propto M_{\star}^{0.30\pm0.11}$ and
$\mathrm{O/H}\propto M_{\star}^{0.32\pm0.09}$. At fixed $M_{\star}$ we find
that, relative to their solar values, O abundances are systematically larger
than Fe abundances (i.e., $\alpha$-enhancement).We estimate an average
enhancement of $\mathrm{(O/Fe)} = 2.65 \pm 0.16 \times \mathrm{(O/Fe)_\odot}$
which appears to be independent of $M_{\star}$. We employ analytic chemical
evolution models to place a constraint on the strength of galactic-level
outflows via the mass-outflow factor ($\eta$). We show that outflow
efficiencies that scale as $\eta \propto M_{\star}^{-0.32}$ can simultaneously
explain the functional form of of the stellar and gas-phase MZR, as well as the
degree of $\alpha$-enhancement at fixed Fe/H. Our results add further evidence
to support a picture in which $\alpha$-enhanced abundance ratios are ubiquitous
in high-redshift star-forming galaxies, as expected for young systems whose
interstellar medium is primarily enriched by core-collapse supernovae.
