Do coiled power cords heat up and cause fires?

While I trust the opinions of those far more versed in electricity than I, I will still pull my power cord ALL the way out, after discovering that mice seem to like to eat the black insulation off the outer cord. I was getting our 5er ready to head to FL last year, and when I pulled out the cord, there were sections where the black sheath was completely gone; and about a dozen places where the cord had been gnawed deep enough that I could see bare copper.