I'm imagining the speaker's face became "flat" as a rubber ball when it bumped right on the ground...is this right?
I found an explanation on a website some time after I posted my question.
Fall flat on my face definition by Babylon?s free dictionary
So when someone says he/she has fallen flat on his/her face, does that person mean both hitting the face on the ground and "making an embarrassing mistake" as the website indicates?