Inuit (Greenlandic)
It is of course possible to derive nouns from verbs using the plethora of derivational suffixes available, but though this is a very integral part of the grammar, it goes beyond the actual verbal system itself.
One possible exception
I just looked at the answer linked to in the question and realised that I’d made this same suggestion two years ago, but had looked it up and partially disproved myself. It seems Greenlandic does have one single entity that could be classified as a non-finite verb form (though, having read the relevant literature again, it’s not entirely clear to me whether it’s really a verb form or a derived noun). This is the suffix -neq, which creates abstract nouns from verbal phrases (note: verbal phrases, not just verbal roots, like most nominalising suffixes).
According to Michael Fortescue’s West Greenlandic (Croom-Helm, 1984, p. 44ff; spelling in Greenlandic forms modified to Standard Greenlandic orthography; glosses modified to better match Leipzig rules), -neq can
convert a clause of any degree of complexity to an NP acting as object or subject of a superordinate verb. […]
- nunaqarfim-mi savaateqarvi-u-su-mi nukappiara-a-lluni sava-leri-neq
settlement.LOC
sheep.herding.place-be-INTR.PART
-LOC
young.boy-be-4s.CONT
sheep-be.occupied.with-NEQ
[…]
The first example savalerineq ‘sheep-herding’ on its own is siffucient as noninal subject. Note the impersonal use of the 4th person contemporative mood in the full sentence […]. The contemporative in these constructions can be said to be non-finite in so far as it de-specifies the subject.
-neq forms do not include markers for mood, person and number from, all of which would be present in the corresponding finite clause, but may be inflected for case and possession as any other nominal form:
The latter marking [for possession] can reintroduce number and person referring to the subject of the clause. If the subject is overtly expressed it will be in the relative case:
- nalu-aa qinnuta-ata qanoq naammassi-neqar-ni-ssa-a
not.know-3s>3s.INDIC
request3sPOSS.REL
how implement-PASS
-NEQ
-FUT
-3sPOSS
There’s a whole lot of fairly complex stuff about how various objects and other constituents are deleted or able to be expressed in roundabout ways using half-transitive constructions, which may or may not be relevant to the categorisation; but including it here would be overkill.
At any rate, it’s clear that, while this is perhaps not a straightforward participle or run-of-the-mill non-finite verb form, it is at least a type of derivation that is unique in Greenlandic in allowing the resultant nominal form to retain certain aspects of its verbal base. As such, it is at least arguably a kind of non-finite verb form.
Goidelic (Irish)