Ions and Ionic Bonds
The variation from ionic to covalent to metallic bonding is not clear-cut, but a matter of degree. For example, the bond between hydrogen and oxygen in water is mostly covalent (sharing electrons), but at room temperature, about one in 10,000,000 bonds break, with the hydrogen losing an electron to oxygen -- the ionic bond you discussed.

However, in the case lithium and beryllium, there is not enough difference between them to favor one getting an electron instead of the other. However, something more interesting happens: both are willing to let electrons go, so there is a sea of electrons, unattached to a particular atom! This is called a metallic bond, and it enables many metals to mingle to form alloys, such as bronze, amalgams and stainless steel.