I had a double mastectomy a week ago. I have swelling under my arms. Should I use an ice pack to relieve the swelling?

Generally ice is not recommended after this type of surgery due to it compromising the blood supply to the skin that is healing.  Discuss this with your surgeon to determine the cause and its correction. 

Most likely this is a fluid collection or perhaps a collection of congealed blood called a hematoma. Neither are dangerous, but can make the site uncomfortable. While your body will take care of the swelling eventually, no matter the cause, you may feel better faster if a liquid portion can be drained. Ask your doctor for an ultrasound of the area to see if this is possible. We frequently perform painless, fast ultrasound guided drainage of fluid collections and can also place drains if needed without going back to the operating room.

Some physicians avoid ice packs for postmastectomy swelling, but I find cold packs in the axilla are comforting, and relieve pain and swelling without compromising blood supply. The breast flaps are more vulnerable to blood supply problems so I wound not use cold on the chest wall. Since your sensation may be abnormal, make sure the packs are not too cold nor kept on the skin for too long - 10 minute intervals are generally safe and soothing. Swelling in the axilla may be due to fluid accumulation which may be aspirated or likely will resolve with time. Are your drains still in? if so they may not be draining the axilla well. If the swelling is associated with pain, a hard mass or redness, or enlarging, consult your doctor immediately to rule out a hematoma (bleeding) or infection. Good luck and take care.