As much as I hate to admit it, she is a woman, and to me it looked like possibly a lipstick tube. She was in flats also, so it wasn’t her heel coming off either, definitely dropped out of her pants leg though.
Also one other thought, I do believe she has a panic button she carries. There was speculation that during the rally she was at she might have pressed it when that guy was asking her a question, setting off the SS response that was filmed a while ago.
Maybe it's a "reset" button like the one she gave the Russian Foreign Minister back in 2009.
You are a dreamer....keep up your hopes, but she is sinking fast.