I find it helpful to draw a distinction between the impulse for suicide and the act itself. There is still a lot of stigma about such thoughts and often people feel a deep shame about thinking of killing themselves. This shame isn’t a protective factor as it leads to further self criticism and low self esteem.
If the therapist is clear that suicide is a mistake and that there are other options, then the client and therapist can go more deeply into the impulse.
In my experience the impulse is almost always a search for peace, an end to suffering and a freedom from the pain that they are experiencing. When the distinction is drawn between the thought and the action these impulses can be acknowledged and even celebrated by the therapist and hopefully by the client. Instead of these impulses being seen as a failing by the client (a common view) they might come to see them as a natural and creative response to a life situation that is painful and imprisoning.
In terms of mindfulness one of the core perspectives is that we are not our thoughts and many people find freedom when deeply realising the truth of this. The suicidal person is often wanting to bring something to an end….mistakenly thinking that life has to end. But when we’re free from the tyranny of thought, self judgement and diminishing core beliefs about self then the suffering caused by the conditioned mind can end without any destruction of the body.