An 85-year-old woman was critically hurt when she was shoved face-first to the sidewalk in a random unprovoked attack near Union Square Park in Manhattan, cops said Tuesday.

Police quickly arrested the 29-year-old woman who allegedly shoved the elderly stranger, who had just stepped off a bus when she was attacked about 12:30 p.m. Monday.

The victim was near her home when she was shoved from behind near Union Square Park East and E. 16th St., police sources said. Her attacker ran off without ever having said a word.

The victim was rushed to Bellevue Hospital with a fractured skull and an “open cranial wound” and underwent  emergency surgery, cops said. She was in critical but stable condition Tuesday.

Cops at the scene arrested Paris Valentine, charging her with assault.

Valentine lives in Flatbush, Brooklyn, according to cops. She was arrested on March 9 and charged with menacing and harassment after she allegedly threatened someone with a Taser in Bedford-Stuyvesant, cops said.

Her arraignment in Manhattan Criminal Court for the Union Square attack was pending Tuesday.