In a previous life I had a lot to do with fabric materials flammability characteristics.
Leather is a good start but beware of decorative treatments, some of which are flammable.
Just about all materials you might choose are able to be treated to make them fire resistant. Try a search along the lines of "fabric fire resistant treatment.".
The problem then is the extent to which any given treatment tolerates laundry processes. Some may need to be reapplied after washing, for instance.
From that previous life and the many flame test procedures I did, some of the then treatments were nothing short of miraculous, transforming very hazardous fabric materials into near flameproof products.