Both are OK on their own. But they also can be used to mean different things.
I fell over the dog. (I fell because the dog was in my way and I didn't react in time to avoid falling. I may have actually fallen on the dog.)
But I fell down the dog doesn't make sense.
On the other hand, I fell down the mountain is OK but I fell over the mountain seems strange.
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(I'm used to North American English.)