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?

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.
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.