This is not true! Actually, the ocean water simply runs off from the flat earth. The water is replaced by rain.
Just to mention that strange concept that, at the edge of a 'flat Earth' you can fall off. Where do you fall into? There must be a gravity force below that flat Earth? Here is an urban legend about flat Earth that has really gone around. In my ancestral village, there was in the late 19th century a man who had traveled to Argentina. There was a rumour that he had been 'at the edge of Earth' . There was no risk of falling off, but the skies touched ground level there, so he could simply use the clouds to wash his hands.
By the way, did you know thet somewhere amidst the ocean there is a buoy floating. It is attached by a chain to a plug, that covers a drain in the ocean floor, some 5000 m deep. Ship captains are warned to avoid the area, since hitting the buoy risks to rip out the plug and, like the emptying of a bathtub, the oceans will empty themselves completely, through the drain into the Earth's core.