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We investigate the ultraviolet (UV) spectral properties of faintLyman-$\alpha$ emitters (LAEs) in the redshift range 2.9<z<4.6 and providematerial to prepare future observations of the faint Universe. We use data fromthe MUSE Hubble Ultra Deep Survey to construct mean rest-frame spectra ofcontinuum-faint (median M$_{UV}$ of -18 and down to M$_{UV}$ of -16), lowstellar mass (median value of $10^{8.4}$ and down to $10^{7}M_{\odot}$) LAEs atredshift z>3. We compute various averaged spectra of LAEs sub-sampled on thebasis of their observational (e.g., Ly$\alpha$ strength, UV magnitude andspectral slope) and physical (e.g., stellar mass and star-formation rate)properties. We search for UV spectral features other than Ly$\alpha$, such ashigher-ionization nebular emission lines and absorption features. Wesuccessfully observe the OIII]1666 and CIII]909 collisionally excited emissionlines and the HeII1640 recombination feature, as well as the resonant CIV1550doublet either in emission or P-Cygni. We compare the observed spectralproperties of the different mean spectra and find the emission lines to varywith the observational and physical properties of the LAEs. In particular, themean spectra of LAEs with larger Ly$\alpha$ equivalent widths, fainter UVmagnitudes, bluer UV spectral slopes and lower stellar masses show thestrongest nebular emission. The line ratios of these lines are similar to thosemeasured in the spectra of local metal-poor galaxies, while their equivalentwidths are weaker compared to the handful of extreme values detected inindividual spectra of z>2 galaxies. This suggests that weak UV features arelikely ubiquitous in high z, low-mass and faint LAEs. We publicly release thestacked spectra as they can serve as empirical templates for the design offuture observations, such as those with the James Webb Space Telescope and theExtremely Large Telescope.

The MUSE Hubble Ultra Deep Field Survey. XV. The mean rest-UV spectra of Ly-alpha emitters at z>3