In the US, this isn't really a common phrase.  The common phrase here is , "I'm dead tired."

Brain dead is a medical term for actually being dead due to a brain injury. 

I hear "Brain dead" used in slang exactly as you describe it, fairly frequently. It gets used much more with the younger population, and is used to jokingly refer to what everyone else is insisting it means. It's not usually used in a professional setting unless you have a really good relationship with that person, and only when talking about yourself as self-deprecating humor. To call someone else brain dead is pretty grievous.

"Sorry, I was brain dead for a moment there"(lost focus and didn't hear what was being said, or couldn't understand something)

"Well I'm brain dead now, so let's pick this up again later." (can't follow directions or remember something)

"I must have been brain dead; how could I have been that dumb / missed that?" (recognizing past mistakes)