Housing authority police say they were investigating complaints of squatters in a public housing complex on Warren Street when they found three people inside.
Two men and a woman climbed out a window on the eighth story.
One man escaped. Another was taken into custody.
The woman who fell landed on scaffolding.
She was taken to Richmond University Medical Center with numerous broken bones.
She was listed in critical condition.
The building is part of the Stapleton Houses, a public housing complex that is the largest on Staten Island.
There are six, 8-story buildings, with 693 apartments housing about 2,148 residents.