... of course they are not.
I've had the pleasure, a while ago, to work with Icelandic wool (for the Bronze Age garments project), and it is a fascinating material. Just like older sheep breeds here in Europe, Icelandic sheep are double-coated, which means they have coarser outer hair and softer, finer underwool. (The Waldschaf is one of the breeds still rare but surviving hereabouts that also has a double coat of similar style.) The Icelandic sheep, just like the Icelandic horse, was brought to the island when it was settled, and like so many things in that area, not much has changed since then.
The difference between the fibres of the outer and inner coat is really astounding, with the inner ones being all fine and soft, and the outer ones very long and strong (and rather coarse). For my project, I only used the inner coat fibres, but I did get a sample of the full fleece to play with, and some of the outer coat taken out. (I haven't done much with it yet apart from admiring it, though.)
If you'd like to read a bit more about Icelandic wool, you can find a blogpost from a small fibre mill in Iceland here, and the same writer explains a bit more about her white colours here. The many different shades of "white" in wools are also something that have fascinated me for quite a while. Just like black is not the same in natural colours, depending on how you dye, with what dyestuffs, and on what (colour) basis, there's a range of whites just like there's a range of blacks. Or, indeed, of any other colour.