"only in the northernmost parts of Norway and European Russia near the Arctic Ocean" is very misinformed. Comrie normally is much better at facts. In Sweden and Norway, Sápmi, Sw. Sameland, extends past the geographical midpoint of Sweden. For example, the southernmost Swedish Sámi "village" (sameby, an official word, among other things implying extensive rights concerning reindeer farming), is located at 62 deg N (Sweden extends from ca 55 to 69 deg N).
It is difficult to give a number of speakers of Sámegiella (cf. Finnish kieli = language), because everybody will be at least bilingual (besides all languages taught in school, there’s probably lots of Swedish Sámi who in addition to Sámegiella and Swedish know Norwegian and/or Finnish). For Sweden, credible numbers for speakers are some 20 000 to 40 000.
Sámegiella, together with four other languages, was rather recently (2000) awarded the status of official historical minority language. This means, among other things, that Sámi persons living in the very northern municipalities of Kiruna (which used to be, and might still be, the largest city in the world, area-wise), Gällivare, Jokkmokk and Arjeplog have a right to get explanations from authoritites in their home language.
The choice of the other four languages is a very interesting thing, politically and linguistically.
They are Finnish, Torne Valley Finnish (Meänkieli), Romani chib (all forms) and—Jiddisch (our spelling).
There have been arguments that Sámegiella should have been split into at least three separate languages, and that Torne Valley Finnish should be accomodated under Finnish. Nobody seems to question the legitimacy of Romani chib.
Quite a few people think that Jiddish doesn’t meet the EU criteria for official historical minority languages, including myself, but OTOH I applaud any effort to protect that interesting language. But it should be a safe bet that Jiddish isn’t the home language of even one person in Sweden. If you stretch "speakers" to include people with only elementary understanding of Jiddish, the number might be towards 5 000.
Regarding the origin of the Sámi people, archaeologists, geneticists and linguists now think that Fenno-Ugric people probably migrated to Fennoscandia from the south, some 6000-8000 years ago, follwing the ice regression. I never heard any theory that they originally might have had another language.
I’m not very interested in DNA comparisons in discussions on language. But I should mention that there’s a unique blood group, A2, very frequently occurring among Sámi people, but not found anywhere else in Europe.