Threw this together over a period of two days, but it took a few weeks of practice beforehand:
Prince of Persia - Completed in 16:29.63 [game time] [2,107KB]
Edit: Now encoded in H.264, saving me a bit more bandwidth.
There's no sound, for two reasons:
1) I played with it turned off so I didn't have to wait for the tune to play at the end of each level.
2) Even if it was turned on, I currently have no means to record it.
So you'll have to do without :)
It's not perfect, but it's certainly no easy task to get a time like that :)
Stuff I missed
# Immediately after starting the first level, hold Ctrl-A [Restart Level]. This eliminates the long pause after landing.
# On level 10, it may be possible to push the second to last guard back out through the gate you enter through instead of wasting time killing him. I haven't been able to do it though, and it might even be quicker to just kill him.
# Also on level 10, it's possible to fall off at the end in such a way that you land right in the corner without enough room to crouch. You still take damage, but the animation and pause is skipped and you land right on your feet.