Even typing this sounds silly and a bit embarrassing, but what the heck!
I used to have this dream at least once a week (since I can remember) until my teen years.
The thing would chase me and I would wake up 0.12 milliseconds before it got to me. I remember being really scared every time, heart beating fast, sweating. Actually, it wasn't really 'me'... it was as if I was a 'presence' and not actual matter.