We can’t exclude it.
Tsakonian is an absurdly archaic variant of Greek, and that speaks to long-term isolation from the rest of the Greek speaking world. It would have to be longer-term isolation than Old Athenian, the cover-term for the enclaves of Greek (Athens, Aegina, Megara, Kyme) blocked off from the rest of the Greek-speaking world by the arrival of ethnic Albanians after the Black Death.
Modern day Tsakonia (the blue bit in this 1890 map) has a coastal bit in Leonidio, but it was cut off from the north through mountains, and the coastal approach required porters to wade out and carry luggage in 1895. You’ll notice that the surrounding areas are purple, not pink (Arvanitika-speaking); the ethnic Albanians were not the reason for Tsakonian being isolated.
We have a few pieces of evidence about Tsakonia formerly. We know from Mazaris’ Journey to Hades, written in the 1400s, that some people in the Peloponnese spoke unintelligible gibberish; earlier on scholars thought that was a reference to Tsakonian, but the surmise is now that it refers to Mani, a (less) archaic variant of Greek spoken in the Middle Finger of the Peloponnese.
And we know that Vatika, at the bottom of the right southern finger of the Peloponnese and near Monemvasia, has almost the same name as Vatka (in Turkish, Misakça), where there was a Tsakonian colony up until 1915. (They were deported to the interior of Anatolia because of WWI, before the 1922 population exchanges.)