Mother has a golf ball size mass in her arm pit hurts her very much and sometimes makes, her arm numb. what could it be?

Possibilities : There are a number of possibilities; it could be an abscess, an infected lymph node, or a tumor pressing on something. She needs to see her dr & have it checked.

Lump under the arm: The under arm lump can be an infection, caused by shaving or a bad bacteria, like staph aureus. It could also be something worse, like a tumor, and in either case, your mother should see a doctor for a proper diagnosis and treatment, especially if it is making her arm numb.