The main record of the Flood is in Genesis 6-9. In Gen. 6:17 God says:
For behold, I will bring a flood of waters upon the earth to destroy all flesh in which is the breath of life under heaven. Everything that is on the earth shall die.
On the face of it, the record reads as if the Flood did cover the whole earth, and many people read Gen. 6-9 in this way. However, many other people read the record of the Flood in a slightly different way because the words and phrases that are used in Gen. 6-9 about the extent of the Flood are used in other parts of the Bible to refer to areas much smaller than the entire globe.
For example, the words in Gen. 6-9 that refer to the earth are sometimes translated as ‘land’ and ‘ground’ in other parts of the Bible and refer to smaller parts of the globe (e.g. Nu. 32:11; Josh. 18:8; Ezek. 32:4; Hag. 1:11).
To take another example, phrases like those in Gen. 6-9 that talk about ‘all flesh…under heaven’ (Ge. 6:17), etc., are used in other parts of the Bible in reference to areas that are smaller than the entire globe. Here are just a few verses:
…the gospel that you heard, which has been proclaimed in all creation under heaven…
The gospel message had not literally been preached in all creation under heaven (it hadn’t by this time, for example, been preached in places like the Americas or Australia).
…this is what was uttered through the prophet Joel:
‘And in the last days it shall be, God declares,
that I will pour out my Spirit on all flesh…
Acts 2 is when the holy spirit comes on the Christians and they start to speak in other languages. In the verses above Peter says that those events were brought about by God pouring his spirit on all flesh. However, ‘all flesh’ here only refers to the Christians, not to everybody on the earth.
So, sometimes in the Bible, phrases like ‘all flesh…under heaven’ are being used in a way that doesn’t refer to the entire world. Two Christian writers sum it up like this:
This [i.e. evidence we’ve just seen] indicates the local stand-point that must be recognised in the understanding of apparently absolute expressions—a thing common to current speech, as when we say of an invited party of friends, “Every one has come,” the “every one” is absolute only within the range of the subject referred to.
Robert Roberts, The Visible Hand of God (Birmingham: CMPA, 1942), 4th Edn, p. 51 [available online here]
We are up against a Hebrew idiom, which can fairly be stated like this: When the Hebrews spoke of “All the peoples of the earth” (or some such phrase) they often meant it in a limited sense. They meant either “All the peoples with whom we have contact”, or “All the peoples with whom God is dealing.”
Alan Hayward, God’s Truth (London: Lakeland, 1973), p. 207 [available online here]
Because of this evidence, some Christians read Gen. 6-9 and see a massive Flood that covered a large area but wasn’t covering the whole globe.
So, it’s possible read the Bible and see the Flood as covering the whole world; and it’s possible to read the Bible and see the Flood as covering only a part of the world.
When we look at the scientific, historical, and anthropological evidence outside of the Bible, however, we see that it tends to support the reading that the Flood covered only part of the globe.