I am not sure of this, but I think that BM Naruto can do so, he can extend chakra arm from where he was hit, then go out of this mode.
Amaterasu flames can't be extinguished, so just moving at extremely high speeds wouldn't do the job if it happened to hit him.
But there's another way. As someone already pointed out he can make chakra arms extend from any part of his body. He can grow a disposable chakra limb from the affected area and that would virtually pull the flames out of his main physical body, then he would just have to discard the chakra limb or turn off his Kyuubi Mode to get rid of the flames.