If a word is in participle form, such as unflagging, but does not appear as a verb (X I unflag; X you unflag; X he unflags), is that word simply referred to as a back formation or is there a more precise term?

the verb 'to unflag' would be a back-formation; but I don't know if that's what you're asking...

Is the word 'unflagging' a back formation? If so, is there a specific term for a back formation in participial form?
Incidentally, thank you.

>Is the word 'unflagging' a back formation?

well, no. (a back-formation from what?)

a ppl is formed by adding -ing to a verb. an example of this is 'flagging'. unflagging is formed by negation, as I suppose you might characterize it; although there may be a grammatic term for this.

The OED notes: When un- is prefixed to present or past participles, these are rarely employed in a true participial function, but become adjectival in character.

e.g.; unflagging energy

then you have the adverb form: unflaggingly pursued
(Sundance: Who are those guys?!)