Silk has tensile good strength but I'm not sure how it does with abrasion. Regardless, it ain't no bitch.
Mongol horsemen would wear a silk vest under their clothes - if they caught an arrow, the silk would wrap around the arrowhead as it entered the wound cavity, actually stopping little bits of fiber, wood, whatever from entering the body and creating infection. Pulling the silk would drag the arrow head out of the wound as well.
In Japan mounted samurai would often wear a silk cape that was arranged in a way so it inflated behind their backs and buffeted back and forth as their horse sped along. Any arrow heading at his back would get entangled or have it's flight interfered with enough that it's path and armor piercing punch were compromised. Sounds silly, but it really worked.
Silk was a big hit with aviators too, it was used for parachutes and harness gear. George Bush, the one who served his commitment, not the retarded son, lived to be president thanks to his chute holding up.
And as someone else mentioned, it's been used in bullet resistant vests (Thailand?)