Can 'fall' be a linking verb?

Yes, it can, but it isn't a linking verb where you use it.

A linking verb is a verb that connects the subject with a subject-complement, which by definition can only be a nominal phrase or adjective.

As it stands:

He has fallen ill.

He fell dead.

And many others are possible, using “fall” as a linking verb.

Like many linking verbs in English that are also used as unaccusative verbs without a complement, this one can only take adjectives as complement it seems, and not nominal phrases, thus:

*He has fallen an ill man.

Does not work, with say “become” or “be”, nominal phrases can also be used.