I think it’s Armenian.

I think it’s most likely a western hemisphere language, either in Mexico/Central America or somewhere in Western South America. No ideas yet to narrow it down though.

Something makes me want to say Basque, but I can’t hear a single Spanish loan or any word-final k’s. Still I think it’s non-IE but European or borderline so. The obvious place to look would be the Caucasus, but I can’t recognize any Russian loans either, and the phoneme set sounds far too reasonable. (Could I possibly be overestimating my ability to recognize loans from languages I don’t know? No, hardly.) Maybe Daydreamer is right and it’s Armenian, perhaps expatriate Armenian. But I’ll go back to Basque.

[Daydreamer, I've been meaning to ask. Are you the same Daydreamer my wife knows from someweb completely different?]

I must admit that the main reason for picking Armenian was the occurence of word final (vowel-ts-vowel) that seems to be a grammatical morpheme in that language. And the lack of Russian loanwords could be a feature of the variety spoken by the vast community of Armenian immigrants to France (i.e. singer Charles Aznavour and the like).
But, the more I write about it the more it becomes clear that I may be completely wrong.
(@Trond Engen: No way. Simon’s site is the only place I use Daydreamer as a nick. My regards to your wife, anyway.)

Here’s a clue: this language is spoken in South America.

I was thinking it sounded Indian of origin (as in India), so I’ll hazard a guess at Sarnami, the Bhojpuri creole spoken in Suriname.

d.m.f.

The answer is Nomatsiguenga (Matsigenka), an Arawakan language spoken in Peru by about 350 people.

The recording comes from the Global Recordings Network.

I actually thought it might be Machiguenga after googling around for the word tasorintsi, which is said several times. Didn’t dare say anything because I cheated ;-)

That’s not cheating, but rather using the information available intelligently.

I wondered if it could be Mapudungún (before looking at the comments above), because I thought I caught a word like Araucana two or three times, and a word like Chile at least once. However, I don’t think the Mapuche use the term Araucania for their land, so I was off on a false trail (even if it did get me to the right continent).

I don’t know what this language is.
One thing I do know: it is not Armenian.