@LittleBear Yes, it depends on whether the wound was deep enough to damage the major vessels in the neck, the jugular vein, and carotid artery. If either of these were cut, She chances would be slim if not nonexistent.
First, to achieve this you will need to not get the character cut around the Trache area, it needs to be closer to the jaw for your character not to die. Closer to the chest if fine but can still be dangerous
Second, How cold is the river because cold it is. The more cold water, more pressure there is
Third, The Carotid is approximately 1.5″ below the surface of the skin, and if severed unconsciousness, will result in death in approximately 5-15 seconds. but it is the purpose for them to survive if they cover it up or if they have Tranexamic acid (Tranexamic acid is given to stop or reduce heavy bleeding. When you bleed, your body forms clots to stop the bleeding.)
Fourth, if it cuts to her Carotid Artery then she is dead. The common carotid artery exhibited the closest average distance to the skin (23.5 +/- 6.9 mm) whereas the internal carotid artery exhibited the closest average distance to the vertebral body (7.36 +/- 3.8 mm, measured to the transverse process).