The knuckles will do it. Because of the way the bones of the knuckles are so close to the surface it can cut the cheekbone. The cheekbone can also cut the knuckles, creating an open wound and causing them to bleed. This is one of the main reasons why you never want to punch out a zombie, if the disease is transferable by blood.
So, no it’s not impossible. She’ll get the cut in addition to a very nice shiner and it will hurt like hell. It’ll swell too, get very puffy, and impede her vision out of that eye for a good long while. It’s also the kind of bruise you keep for a few weeks. So, she’ll be feeling the sting (haha) of failure for a while.
The real question here is: why are they allowing blows to the head? Why are their superiors allowing them to spar without any hand protection? This is the kind of injury you get in an actual fight, not really in a sparring match.
When you spar, you take steps to prevent these sorts of injuries from happening. Contrary to popular fiction, sparring is about testing your abilities within a controlled environment and doesn’t actually involve any intention of hurting your partner. No one should be going home with serious bruises. If they are, then it’s a failure on the part of everyone involved. Sparring is a training exercise and training is meant to teach, not break.
This isn’t to say one doesn’t get bruises when you’re sparring. You do. However, the injuries are very minor. Muscle aches, maybe a few small bruises that’ll stick around for three weeks, but not bruised ribs, pulled ligaments, cut knuckles, or any injury that leaves you seriously impaired. An impaired student is unable to train, if they are unable to train then they fall behind and might end up blowing their physical conditioning on the period it takes for recovery. You also negate the chance of greater injury with strict rules that reduce the chance of lesser injury.
There are certain sparring tests which are what we might conveniently term “balls out” such as graduation exams within certain special forces branches of the military, but those are singular instance and, again, under extremely vigilant supervision.
Too often, this sort of scene happens in fiction without any consequences. The assumption is that with “hardcore training” there are obviously going to be injuries. This is just not true. The more dangerous the training, the more precautions are taken because, again, you still want them present and learning tomorrow. This isn’t just a case of one student being at fault, both are, because the intensity has been pushed to a point where this occurs. The teacher as well if present (and especially if not).
I mean, if they’re not wearing any hand protection then I’m going guess they also aren’t using mouth guards (or cups if one of them is male). In which case, the hit to the face also risked: chipping teeth, losing teeth, cutting the inside of the mouth, splitting the lip, and biting off a piece of your own tongue. Not to mention the prospect of a concussion. Remember, it’s all fun and games until you’ve got to explain to your teacher why your mouth is full of blood.
So, while yes, your friend is right, she’s also wrong. Training accidents do happen, so these are just some things to think about.