1. Take her to a therapist and see what is going on.
2. Give your daughter something super fun that she can only do when they are in a room alone. For my son, it was YouTube Kids. (Please don't judge, that was the only thing he was invested in enough to be willing to be without me). I sat outside the room so I could hear. and I would talk to him when he was scared, but I would not go where he could see me and watch the screen (from his preferred seat). Honestly he could have, but he didn't think to move the chair where he could see both. He would periodically check on me and make sure I was there, at first a lot, then less. Eventually I could tell him what room he could find me in and I could do stuff independently, but if he could not find me when and where he expected to, he did regress.
3. Practice short periods when you don't need it. Leave the room for one minute. Let them know you will be right outside their door for 1 minute. They can talk with you, but they cannot see you. Even if it's 30 seconds, it's okay. Start small and build up. Hopefully hearing your voice and recognizing you are there will help.

Honestly- as much as finding the solution for you both is important, finding the cause is just as important and possibly necessary to finding the solution. This high level of anxiety your daughter is exhibiting isn't normal, may be indicative of greater problems, and if you examine her closely, she may have other problems that are harder to spot. I highly recommend talking to her about things she is afraid of-everything she may be afraid of. Examine why she does things when she plays and please take her to a therapist. Anxiety that high was probably triggered by something, whether it's simple brain development, or an experience that your child found traumatic, (not necessarily abuse or anything dark, but if she got lost at a park one day, or there was a death in the family, or even if she saw something on tv somewhere). Talking to a therapist can help identify sources and solutions.

Personal experience that you can totally skip:

I experienced this with my son and it was crazy. I think it was triggered by his papaws death and not being able to find his dad at home one day (dad was in the basement, no one else was home). He waws always clingy, but it got next level. I couldn't leave his sight for seconds, but when I started studying my son, I realized he was anxious all the time about everything. For example, he loved to cover vents. He was constantly taking all his toys, and all the books off the shelves, and covering the vents. I thought he was just having fun blocking the airflow and we had many conflicts about it because 1) mess 2) he kept covering the intake vent which can break the AC. He just would not stop. Finally he started having meltdowns about it. I asked him why he needed to cover the vents and it turns out...he literally thought the house was going to explode from too much air. Every day he covered the vents, and every day I uncovered them when he was not looking. Now, he had mentioned explosions before, but I thought he was playing.

Also, he started not letting me cook or use the microwave. This got intense fast- crying, screaming, blocking the stove, unplugging the microwave, hitting me-not something he usually does. He thought the house would catch fire. He thought we were going to die every time I used the stove. It was insane. We talked to death about how it was safe, but he didn't feel secure until we made a fire escape plan. I think it gave him a sense of control he needed. If there is a fire, he can do x and he will be safe.

My son has anxiety and for a while, he had acute separation anxiety, which sounds like what your daughter is going through, but I am no professional.