We have observed, with high spectral resolution, the BeII resonance doublet at λ 3131A in five G and K-type main-sequence stars belonging to the Hyades open cluster, as well as in four K-type probable members of the Ursa Major Group. We carried out a detailed abundance analysis via spectral synthesis fitting and studied the utility of the BeII lines for deriving beryllium abundances in late-type dwarf stars. Reliable abundances were derived for three Hyades stars with effective temperatures in the range 5700-5200K. For stars cooler than ~5200K the observed feature in the position of the weaker BeII line is strongly affected by a line of other element which we tentatively identify as Mni λ 3131.037A. Due to this fact, we could establish only upper limits to the abundances in these very cool stars. Lithium and beryllium abundances in Hyades stars with effective temperatures in the range 6000-5200K show a complete different behaviour. While lithium is rapidly depleted towards decreasing temperatures, reaching a relative difference of ~3 orders of magnitude, beryllium remains fairly constant for the same stars, with an abundance similar to that exhibited by hotter stars of the cluster and close to the solar value. Standard evolutionary models cannot account for the present Li and Be observations in Hyades late-type stars. Rotational mixing associated with angular momentum loss can explain the observed abundances if the initial angular momenta of the stars were progressively larger for decreasing stellar mass. On the other hand, mixing induced by internal gravity waves satisfies the observational constraints.