This tactic is an ugly one and can cause some serious psychological damage. Most victims of narcissistic abuse are highly sensitive, empathetic people and in this circumstance, their empathy is weaponized for coercive control. You will see this in times of desperation, particularly in relationships where the abused is fed up and ready to leave. The subtext in these situations is that only you staying will keep them alive, which places the entire burden of their lives on the victim. Due to trauma bonding and empathy, the victim stays in order to prevent the suicide, feeling more and more stuck in a relationship.
Whether implied or specifically stated, calling for help is considered a huge betrayal. The victim feels like they are solely responsible to save the abuser. Calling the cops would ruin their reputation, or they might state that going to a mental hospital would ruin their whole life. Calling for help then means the victim then feels the weight of responsibility of protecting the abuser's reputation, career, legal status, etc.
Even as a mental health professional, I would never manage suicidality alone, and I do not expect you to either. If someone was in my office (or digitally) and states they are having active thoughts and a plan of suicide, I am ethically obligated to call for assistance immediately. You will notice first responders will not arrive without a partner to support them.
If someone states any threat of suicide, call 911. If they are serious and want to harm themselves, help is already on the way and this often starts a chain reaction in getting them help if they need it. If this is only a ploy to get you under control, the abuser will learn that this is not a way to control you without authorities getting involved, which they do not want. This indicates to the abuser that this is not something you will try to manage on your own, and hopefully removes it from their playbook. If authorities have arrived, and this person is now hostile because you called the police, help is already with you.
"If you leave me, I'll swallow this whole bottle of pills and it will be all your fault!"
"Well I guess I might as well shoot myself since I have nothing to live for once you leave!"
"If you kick me out, I will slit my wrists and then everyone will know that you abandoned me and that led to me killing myself!"
Someone using this tactic against you, elevates your overall risk in this relationship, and you need to consider safety planning and support to exit the relationship, up to and including a domestic violence shelter.
Learning about the tactics used in narcissistic abuse is a strong way to begin to notice ongoing patterns of abuse in your life, however this blog is not intended to replace therapy. If you determine that there might be narcissistic abuse in your life, finding a therapist who understands narcissistic abuse is highly recommended.