Yes. All mattresses sold in the US must be flame retardant by law. Most materials used in mattresses, such as foam, latex, and wool, do not require to be treated with any chemicals to comply with the flammability standards.
The only component that needs to be treated with flame retardants is cotton batting. Manufacturers use odorless and natural boric acid (aka borate, aka borax) as a fire retardant. Boric acid is not a known carcinogen by the EPA, the state of California, or the IARC International Agency for Research on Cancer. For most mattresses, manufacturers also use fire retardant on the mattress cover fabric as a fire barrier. Another added benefit of boric acid in mattresses is the bed bug and dust mite factor.
If you want to avoid all fire retardant chemicals, consider Natural/Green or Organic mattresses made with wool. Wool is a natural fire retardant. It also regulates your body temperature when you sleep. Mattresses made with wool meet flammability standards and don't need to be treated with boric acid.
You can also purchase certain mattresses without fire retardants with a doctor’s prescription from a licensed health care professional.