How can the solid state structure of cyclopentadienyllithium be explained?
This summary identifies several calculations indicating that covalent interactions between lithium and cyclopentadienyl moieties do occur; the interaction is mostly but not purely ionic.

Draw the occupied $\pi$ orbitals of the cyclopentadienyl groups and, for comparison, draw the $s, p_x, p_y, p_z$ orbitals on lithium with the $z$ axis along the length of the chain.  Observe:

• The lithium $s$ and $p_z$ orbitals overlap the lowest energy cyclopentadienyl $\pi$ orbitals.

• The lithium $p_x$ and $p_y$ orbitals overlap the bonding degenerate pair on the cyclopentadienyl ring.

With these overlaps, we can construct three lithium-cyclopentadienyl bonding combinations per cyclopendienyl group, one corresponding to each cyclopentadienyl $\pi$ bonding orbital.  We then occupy these orbitals with six electrons per ring.  In effect we have extended the aromatic coupling of the cyclopentadienyl ion into the third dimension with the lithium orbitals.

Heavier alkali metals do not work as well because their orbitals are too diffuse, but they could overlap larger rings.