They'll not only continue to grow, but actually become even tastier.
The cold weather enhances their flavor. I live in the U.S. so I have no way of knowing how cold your temperatures may get, but it doesn't hurt them to be left in the ground even when frozen. They'll be just fine in the spring.
If you have enough to store, try layering them in a plastic container in common play ground type sand. Make sure it's slightly damp and put a shallow layer of sand in followed by successive layers of sand and carrots. It helps if you can put a tight lid on them to keep the sand from drying. We've eaten really good carrots this way for years. Just keep them in the basement or garage where it's cool and dark. (Later in the winter, they may tend to split slightly... be sure to rinse them if they do 'cause they get a little gritty).