This question naturally arises from reading Feynman Lectures Vol III 14-3 The Hall effect, online available here, where Feynman states the following:
The original discovery of the anomalous sign of the potential difference in the Hall effect was made in a metal rather than a semiconductor. It had been assumed that in metals the conduction was always by electron; however, it was found out that for beryllium the potential difference had the wrong sign. It is now understood that in metals as well as in semiconductors it is possible, in certain circumstances, that the “objects” responsible for the conduction are holes. Although it is ultimately the electrons in the crystal which do the moving, nevertheless, the relationship of the momentum and the energy, and the response to external fields is exactly what one would expect for an electric current carried by positive particles.
I do understand how the hall effect suggests positive charge carriers, you may also compare this question and its very good answers about the behavior of holes in magnetic fields for clarification.
However, beryllium is a metal and more importantly not a semiconductor, thus (1) there is no obvious significance of the valence band and (2) the concepts of dispersion relation and effective mass are unclear to me (as this is a metal). How can one explain the Hall effect suggesting positive charge carriers in beryllium considering it is a metal?
I did search for papers and also general information about beryllium, but I was even unable to confirm the statement that beryllium shows reverse polarity in hall effect. I also did not find any other comment on the charge carriers being positive.
Edited based on a comment which may make less sense now without original context. The comment made me think that me imagining electrons in a metal as a free electron gas may be what I'm oversimplyfing here. Is thinking of the electrons in a metal as a gas under certain constraints more appropriate and necessary to explain this?