Can you die from a small self inflicted injury?

Yes,: Although many people injure themselves on purpose or accidentally without last harm, self injurious behaviors (sib) can produce lasting damage and even death, depending on the kind of injury--severe head banging, for example. More important, perhaps, is to understand the need to hurt oneself, because usually is a symptom of significant emotional distress of some kind.

Depends on Type: It would be very rare for a "small" injury to cause death. However, if the injury is in the right (or wrong) place, this could occur. Also, frequent small injuries could lead to increased risk of infection, which could become serious.