Is martial arts a performing art?

That depends. A martial art in the strict sense is a fighting art, not a performing art. And martial arts performed in fighting competition are not a performing art either: they are a sport. But when when martial-art drills are performed as a display (e.g. forms performed in ceremonies and parades) that is a performing art, and when choreographed "fights" are staged as part of a dramatic performance (e.g. for films or in Chinese opera), that is a performing art too, especially when showy acrobatic moves instead of ones that could be effective.