There isn't really evidence (outside of the Bible) of a flood that covered the WHOLE WORLD, including the tops of the Himalayas et al) under miles of water (nor is there enough water in the world to do it, unless the oceans were a LOT shallower than they are). But there certainly IS evidence of a lot of heavy flooding all over the world in the same time period, perhaps 10,000, perhaps 15,000 years ago.
There are whole cities underwater on the submerged plain that links Indonesia with Australia and the mainland.
The legends of the eastern woodland Indians include the tale of a great flood, particularly which broke open Long Island Sound and made it part of the ocean.
West coast Indians have similar stories.
Actually, there's a flood legend of one sort or other in all of the world, including the European legend of Atlantis.
Because there really was a flood.
Its source was more mundane, and the effects were not QUITE as dire as the drowning of the ENTIRE world, but they were pretty dire.
The waters never receded from this flood.
It's why the Siberian Land Bridge is still underwater.
The cause: global warming.
In the last ice age, a lot more of the Earth's water was in ice, and a lot more of the Earth was covered in great sheets of ice. Then the climate heated up and that ice melted. Not slowly. The Great Lakes are the inland remains of that ice sheet over America. But when the oceanic ice melted, it raised the level of the oceans. Probably not slowly.
Those were disastrous times. Weather pounding, coastal lands submerging, islands (like Britain), being sundered from the mainland. People used to live on those drowned continental shelves, and they are all around the world. Every culture has a distant myth, a memory in legend of its own Atlantis experience. Because when the Ice Age ended, men were there, and the oceans rose by hundreds of miles.
It probably didn't happen in 40 days, but it probably didn't happen over 4000 years either. Every coastal culture's legends remember "The Flood", or "A Flood", back at the dawn of time. The Mesopotamian culture of Sumeria, whence Abraham came, remembered it too. "Eden" is at the bottom of the Persian Gulf.
The "WHOLE WORLD" to the people who wrote Genesis was a whole lot smaller than the Whole World - They hadn't an inkling about most of the other continants, AFter all, if you sailed too far to sea, you would fall off the edge...there was no North/Central/South America to them, nor Austrailia - as far as they were concerned. They probably had no idea there was a land that would become Englang...etc
In other words, their WHOLE WORLD could flood without it affecting the Whole World
Global warming. Bush's fault!
As a side note: today, something like 80% of the world's population lives within 200 ft of sea level...