Re: Tsunade stated that she can't die in a fight
It's useless to discuss that. While I'm well aware Tsunade is not immortal and that she would die if beheaded in her default form, the full extent of her healing factor during the activation of Byakugou is unknown. We know it works perfectly fine with impalements, but getting an entire body part tore out of your body is much more serious than getting pierced and while I don't think she would be able to grow a new limb back, there's just not enough material to confirm or deny the possibility. We can only guess.
I would like to highlight, however, getting your lower half chopped from the rest of your body and surviving is possible even in real life since there are no vital organs in the lower half of the human body. Of course the possibility of death by blood loss in incredibly likely, but it is possible to survive that if the person in question gets emergency medical treatment. So no, you are not immortal for surviving being cut in half, just a really tough person.
I think I made my point clear, but now I must go to her character's defense as well. Remember when I mentioned her insta-healing mode with unknown limits? While it is pretty obvious Tsunade would just die if beheaded in her base form, since she wouldn't live to heal herself from such a wound, what about getting beheaded during Byakugou?
I've seen people arguing it wouldn't be possible from a scientific standpoint, but it is a manga with lots of fantasy involved. From a scientific standpoint, even with supposed immortality there would be no reason for Hidan to talk with his head severed because his vocal cords would be gone and he would be struggling to breath as well, and any attempt of teleportation would kill the user for the sheer fact air has mass and you would be materiazling your body within it. A fantasy work recquires some acceptable breaks from reality, so it is useless to use scientific logic to argue against growing you head back during an insta-healing mode.