It is possible that you have a suit here. It depends on the extent of the rash, whether you can prove the cause of the rash, and a number of other factors. I recommend you contact a lawyer to discuss. If you'd like, you can call 312.878.2357 for a free, private consultation.
You never know what a lawyer can get you. A war story for you: I had a client who used a hair dye for beards which caused hypopigmentation to his face, and the CEO flew to my office and offered 200k. Thus, you never know. Also, save the product.
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Just because a product has been around for decades does not mean impurities or worse can not creep in to a batch. Have we forgotten the Tylenol scare? The issue in your case is causation. Someone needs to test that container of deodorant and write a clear narrative.
Here is what causation is about: [Blue Link Below]
Law Offices of Andrew D. Myers, North Andover, MA & Derry, NH provide answers for informational purposes only. Actual legal advice can only be given by an attorney licensed in your jurisdiction, thoroughly familiar with the area of the law in which your concern lies. This creates no attorney-client relationship.
When I read the title, my immediate reaction was "use a different deodorant." I've personally used a deodorant I had an allergy to, developed a light rash, switched deodorants, and all was well.
But if your doctor says its a third degree burn, I think my initial reaction was too dismissive. As one of the other commentators advised, save the product. Also keep careful records/notes of when you started to have symptoms, and track any changes in your symptoms. Get photographs of the affected area, pay particular attention to getting good lighting. Take some photographs with a ruler to show the scale of the injury. Photograph any changes in the affected area. Obtain a copy of your medical records, and be sure your doctor actually said it was a third degree burn.
When they hear "I have a rash from my deodorant", any lawyer is going to question your damages. So when you call a local lawyer, if they are initially dismissive, explain what your doctor said and send them your photographs and a copy of your medical records. This may help you obtain counsel. It sounds like this may be more substantial than a simple rash. And as one of the other commentators noted, sometimes products are defective or compromised.