Short answer: yes they do.
A firm called Decode Genetics assembled a genetic map of the entire Icelandic population a few years back, since the family relationships of every single native Icelander are well documented and they purchased enough genetic information to generate DNA profiles for every Icelander.
There are about 300,000 Icelanders and they are all descended from the well-documented 40,000 person Icelandic population of 1800.
That group of 40,000 are descended from a group of probably no more than 4 or 5,000 who survived the Second Plague of 1495.
Pretty much every Icelander can track their complete family tree to at least that date.
wow, interesting.
Icelandic people are fascinatingly odd.