Though the term "splitting" is misused often by line staff, it is indeed a symptom of Borderline Personality Disorder. In the confusion over the boundaries of what is BPD, what is PTSD, and whether someone has a full Personality Disorder or simply traits, lines are hard to draw. All this granted. But splitting, the psychiatric symptom, does indeed lead to "splits" in the staff of how a patient should be handled at any given point. Some are moved to rescue, some to punish. Default opinions about What Should Be Done are frequently appealed to, with staff slowly being maneuvered into acting out the patient's split - their either-or thinking by engaging in unlistening, either-or behavior themselves.
But it is not the patient's actions which create the split amongst the staff. Those maneuverings and attempts to externalise inner conflict into the actions of those around reveal staff conflict, not create it.
This diagnosis-specific knowledge has more general application to all "difficult" people, such as the ones you work with or live with. You might examine whether they are creating the conflict in the office, the church, the neighborhood, the family, or are they simply revealing a split that is there anyway?