Is a 3 km wide map OK for my first open-world game?

that depends on how big your team is, how much land vs. water your islands will have, and how much content density your game needs (i.e. survival games often have a lot of empty space while RPGs need much more stuff populating the world)

consider the land of Skyrim where the heightmap is 6.8 x 5.4 km

from that, only around 5 x 3.5 km are playable. and still it has an enormous amount of content in that amount of land