lol I think you're suppose to prevent that from happening in the first place by using Chapstick or something similar. At least that's what I do to keep myself from doing the same thing. It hasn't stopped it, but it has cut down on it.
I use vaseline on my lips, and any other part of my body that has really dry skin. I have had the exact same problem, Jackie-San, and I know that one thing that really feeds the problem is lip-licking. It takes a lot of concentration and will power, but you need to stop licking your lips all the time. Get a tiny jar of vaseline, and keep it in your purse.
When you have the pieces of skin hanging off, put a liberal amount of vaseline on your lips, and use your finger tip to rub it in, gently, back and forth. As your lips get warm and lubricated, the pieces of dead skin will come off gradually under your finger tip. Don't pull off the ones that are firmly attached to live skin. Clip them off carefully with small scissors.
Do this every time you feel dead skin accumulating along the outsides of your lips, long before the pieces get big. I do this almost every day. Eventually, you will reach the place where you are preventing the pieces of dead skin hanging off, and that's a great place to be. :-)
Exfoliate your lips! It works great for me. Very gently brush your lips when brushing your teeth in the morn or eve, pref with a SOFT bristle toothbrush and only after the skin has been well saturated with warm water. Leave a bit of toothpaste on the brush then brush lightly in a circular motion. Immediately pat skin dry and apply your favorite lip balm. Massage it in well. I suggest Burt's Bees or Blistex. I would use Carmex if lips are prone to excessive dryness and blistering or if your lips are constantly exposed to the sun. Several lip balms contain an SPF. If you are into SPF, as I am, I suggest applying it liberally and often throughout the day. Do NOT brush lips in areas where skin is broken or it will aggravate the area and make it worse.