Yes, certainly. The human body has several systems which are separated from each other such that fluids cannot freely move between them, and the pressure within each is maintained in a very precise manner.
Examples which come to mind are the cardiovascular system (which has variable pressure all the time, but it also varies in a strictly controlled way), the eye, the inner ear, and the brain ventricles and other parts of the nervous system which are bathed by the cerebrospinal fluid. Whenever the pressure of one of those is disregulated, the health consequences are severe: hypertension, glaucoma, endolymphatic hydrops or intercranial hypertension respectively. This list is not comprehensive, it just gives a few examples for better understanding.
When you have a small wound, the vascular system gets opened, but there is very little blood flow through the peripheral vessels affected, and the body clots it quickly enough that it does not bleed out. If you get a large tear in a major blood vessel, then the blood pressure can indeed lead the body to bleed out.
If you get a large wound directly into the eye, pleural cavity, cranium or labyrinth(!), a pressure differential is not your main problem.