Combine chicken breasts, chicken broth, cream of chicken soup, cream cheese, and dressing mix in a multi-functional pressure cooker (such as Instant Pot®). Close and lock the lid. Seal vent. Select Poultry or Meat setting according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.

Release pressure naturally according to manufacturer's instructions, about 5 minutes. Release any remaining pressure by opening the vent. Unlock and remove the lid. Transfer chicken to a cutting board and shred using 2 forks. Mix back into the sauce.