Brittany is a bigger example of flags depicting a heraldic fur. See:
The flag of Brittany depicts nine stripes, representing the nine historical parishes historically used to subdivided the stripes; the five black stripes represent the five parishes where French is spoken, and white represent the four parishes where Breton is spoken. On the superior left canton, we can see a rather unusual pattern: it's a fur!
There are basically two main heraldic furs, being the others mostly variants of them: the ermine (present on the flag above) represents the pelage of the animal of same name, while the vair (rarer in flags) represents the snow pelage of the Eurasian red squirrel.
Noble furs were always very expensive products, so having a fur on your shield, on the Middle Ages, represented mainly nobility. The Dukes of Brittany have, as their coat of arms, a plain ermine field! During the history, the ermine pattern became highly associated with Brittany, being presented in the symbols of many towns of the region.
Although I think the concept of flags made of fur as odd, it's very legitimate that they can be transposed from shields to cloth. After all, sometimes, like above, it generates really charming flags!
Comments are welcome!