It is certainly possible that you are having an local contact dermatitis or allergic reaction. It would be quite unusual for that kind of reaction to develop simply from handling the medication, but technically possible. Another possibility would be that you came in contact with some other unrelated substance that is causing the peeling.
The other possibility would be dyshidrotic eczema
. This is a type of skin inflammation in which small blisters form under the skin and lead to itching, redness, and peeling of the palms of the hands or soles of the feet. This would be more likely if you have a history already of sensitive skin, eczema, or skin dryness in the past. Dyshidrotic eczema is treated by making sure to avoid irritating the skin more (for example, wear gloves when washing dishes), keeping the skin very well moisturized with a thick "greasy" ointment, and applying an over the counter topical steroid cream to the most stubborn areas of redness, itching, and peeling. Avoid scratching or picking as this can make the irritation worse and may lead to infection.
If the condition does not clear up soon, you should talk to your primary care doctor
who can help you figure out a treatment plan.