I would not be driving in the mountains before I found out what that brake light is. Many years ago I was coming up on a mountain pass on a Sunday, in winter with snow when my brake light came on. I stopped at a motel. It turned out it was just moisture in an electrical connection and had no connection with the brake operation, but I was gad I did not go on.
Yours may be little or nothing or it may be a serious safety problem.
Assuming you have the owner's manual it should tell you want that blinking light means. I suggest you check it. It would also help if you could tell us what make model and year car you have.
OOps I better get my eyes checked I read that as “Brake light flashing.”