Was there really a flood? And was it just a "local" flood?

The really short: yes to the first and no to the second.

A world-wide flood really happened according to the Bible.

Some say that just a local flood occurred, but then the promise that God wouldn't do the same again (hence the covenant reminder of a rainbow) wouldn't be true because there have been many localised floods causing widespread death since then.

Rather than try to twist the interpretation (or worse the scripture) in a complicated manner to match a localised flooding for Noah, why not just take the plain understanding of a worldwide flood?

"12 And God said, “This is the sign of the covenant I am making between me and you and every living creature with you, a covenant for all generations to come: 13 I have set my rainbow in the clouds, and it will be the sign of the covenant between me and the earth. 14 Whenever I bring clouds over the earth and the rainbow appears in the clouds, 15 I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become a flood to destroy all life. 16 Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth.” " - Genesis 9:12-16 NIV