
Sounds oh so familiar! I was at that stage with my Border Collie pup for what seemed like for ever, but of course wasn't! Don't worry, she got it in the end and so will yours. She's now 10 months old and goes down, very quickly, at distance, without a treat and stays there until released.
I find that a lively game (lively for you as well, I'm afraid!) of "see who can get down quickest" works a treat (without a treat for you or the dog! lol).
At first on the lead, favourite toy in hand, run a few steps and then drop to the ground facing your pup. Say "down" too. She should go down with you, so she gets the toy for a few seconds and lots of praise, then off you go again. Keep it fun and short and add in changes of direction and vary and length of time before you drop - from quickly, to after a few moments and back to quickly again - keep her/him guessing. Once she/he understands the game you can do it off the lead and at more of a distance and will no longer have to throw yourself to the ground too!
Good luck and let us know how you get on.