If there was a woman injured in a traffic accident, am I permitted to carry her to hospital?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

You are required to aid that woman by carrying her to the hospital if no Mahram (permanently unmarriageable) men of her are available to do this. By doing so, you save an inviolable soul. Allaah Almighty says (what means): {Because of that, We decreed upon the Children of Israel that whoever kills a soul unless for a soul or for corruption [done] in the land – it is as if he had slain mankind entirely. And whoever saves one – it is as if he had saved mankind entirely.} [Quran 5: 32]

Being in seclusion with her should not prevent you from aiding her because there is a greater evil, which is her death. It is one of the Islamic principles that when there are two evils, the lesser should be done in order to ward off the greater one. Shaykh Ahmad Al-Hamawi a Hanafi scholar, said: "When there are two evils, the greater should be avoided by doing the lesser." 

Allaah Knows best.