Common problem if the mitt hasn't been looked after properly. Proper lambswool mitts need to be washed and dried properly or they disintegrate.
Don't use a washing machine or tumble dryer. After washing the car, rinse the mitt, gently wring the excess water out and then let it dry naturally - like normal sheep dry! The internal "glove" needs to be dried or it will turn rotten and disintegrate too.
Personally, after detailing the car, I rinse my mitt in the shower, then hang it using a coat hanger in the bathroom. The hanger is a metal wire one that's shaped so it opens up the inside of the mitt (to allow air to reach it to dry) without stretching the wrist elastic. Other people have used a plastic bottle cut out to fit inside the glove to keep the inside open and aerated.