Abandonment issues, by themselves, are not a diagnosable mental illness, though when someone is unable to deal with their abandonment issues in a healthy way, they can lead to a mental illness.
Abandonment issues are a classic symptom of borderline personality disorder (BPD), a mental illness that is known for extreme mood swings, chronic feelings of emptiness, relationship problems, and an unstable sense of self.
Abandonment issues can also be a form of anxiety, which may result in the person being a "people pleaser" or engaging in codependent behaviors in an effort to lessen their chances of being abandoned by a friend or romantic partner.
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