Is there any sunlight on the dark, far side of the moon?

Yes, but only during the day. There is no “dark side” of the moon. That term is metaphorical, meaning unknown, like the phrase “the dark ages” which was coined by a historian about an era about which little was known at the time, not because it was literally or figuratively dark. read more

Both sides of the Moon experience two weeks of sunlight followed by two weeks of night; the far side is sometimes called the "dark side of the Moon," meaning unseen rather than lack of light. About 18% of the far side is occasionally visible from Earth due to libration. read more

Interestingly, "new Moon," which is the time when the Sun and Moon are on the same side of Earth, is when the face we see from Earth actually IS dark. So, calling the part that faces away from us as the "dark side" really IS a mistake. read more