Mix a bit of honey and baking soda, use it to gently rub the lips. This will remove dead skin cells and the peeling skin. Then the lips will become very soft.
You can also use a toothbrush to gently rub your lips. This will remove dead flakes and make your lips soft and supple. After this, apply a lip balm.
You can also use honey. If your lips are severely chapped, it might cause a tingling sensation. Honey is the best natural antiseptic and cleanser, it won't harm the skin.
If there are mouth corner cracks, it usually indicates a deficiency of Vitamin B2. You will need to take vitamins supplements.