  Abundances of beryllium in metal-poor stars scale linearly with metallicity
down to [Fe/H] ~ -3.0. In the stars where Be has been detected at this
extremely metal-poor regime, an increased abundance scatter has been previously
reported. This scatter could indicate a flattening of the relation between Be
and [Fe/H]. Our aim is to perform a new investigation of Be abundances in
extremely metal-poor stars and try to clarify whether a Be abundance plateau
exists. We revisited the Be abundances in a sample of nine dwarfs with [Fe/H] ~
-3.0. Additionally, we analysed the Be lines in the spectra of stars BPS BS
16968-0061 and CD-33 1173 for the first time. We took advantage of Gaia DR2
parallaxes to refine values of the surface gravity of the stars. Abundances of
Be were determined using spectrum synthesis. Some of the stars indeed suggest a
flattening. Between [Fe/H] ~ -2.70 and -3.26, the Be abundances stay mostly
constant at log(Be/H) ~ -13.2 dex. Nevertheless, for several stars we could
only place upper limits that are below that level. Most of the sample stars are
consistent with having been formed at the progenitor of the so-called
Gaia-Enceladus merger. Two out of the three stars likely formed in-situ are the
ones that deviate the most from the linear relation. The mixed origin of these
stars offers a clue to understanding the flattening. We suggest that our
observations can be understood as a consequence of the inhomogeneous star
forming conditions in the early Galaxy. Without efficient mixing, the early ISM
would be characterised by a large scatter in Fe abundances at a given moment.
Beryllium, on the other hand, because of its origins in cosmic-ray spallation,
would have more homogeneous abundances (in a Galaxy-wide sense). We therefore
suggest that the observed flattening of the Be-versus-Fe relation reflects a
stronger scatter in the Galactic Fe abundances at a given age [abridged].
