Tatar-dominated Siberia (latest POD maybe the 14th-15th century with the Golden Horde). Like OTL the circumpolar peoples remain animist for centuries, but Islam gradually makes inroads by the late 19th century and by the late 20th century they are entirely Muslim albeit largely practicing a sort of folk Islam.
There were a handful of Muslim pirate raids on Iceland in, I believe, the 17th century. Maybe they decide to stick around (though Lord knows why) and Denmark lacks the wherewithal to kick them out.
It's a stretch, I know.
ETA: Sorry, looks like my scenario goes against the OP's ground rules.
The same way Muslims living in "the lands of the midnight sun" do now, or possibly some even more
pragmatic way since we are talking natives rather than imigrants.
You could keep the Khanate of Sibir around, alternatively you could create an outward facing Islamic state in East Asia that would go explore Siberia and Alaska.
You could keep the Khanate of Sibir around, alternatively you could create an outward facing Islamic state in East Asia that would go explore Siberia and Alaska.
