It doesn't really matter.
This hasn't really been addressed in other question, while they have provided plenty of options.
What I mean is that any advancement's will be mirrored by any naval power in your world. Or at least should be. So even if you have only inefficient oars, probably everyone will have them. If you have Golmn powered oars, everyone will have them. That's of course assuming the windless-ness will affect everyone on equal terms.
We'll assume by this:
Imagine now a naval power that inhabits an island that is definitely not windy
Means that areas away from the island are normal. And that the lack of technology makes sailing still a viable method of propulsion outside the island.
Now, imagine attacking the Island nation. Would you use sail only ship? Probably not right, so you would use whatever is currently the best option. If you went to attack them and you had oar ships and they had paddle boats and whopped the tar out of your navy, the next time you would probably have your own paddle boats. If the Islanders had more efficient hull designs, you would see that and quickly copy the design.
Basically it would be an arms race. Well, assuming the island was valuable enough. Otherwise you would just avoid them.
From both a defense and offense point of view, there some considerations to make.
- Outsiders would need dual propulsion ships to attack.
- Islanders would need dual propulsion ships ships to attack
- Outsiders could defend with only sail powered ships
- Islanders could defend with only non-sail powered ships
Ships that have to be both sail and non-sail are at a disadvantage to those that are just one propulsion method. Generally there would be less space for guns and less space for fighting men. You wouldn't send your oars men ashore in an attack because if they got killed you whole navy is lost.
I am assuming too that sailing from the island to anther land mass would be more efficient by sail then by oar ( for instance )
Otherwise I really like some of the other Ideas quite well.
My preference would be something like a hamster-wheal powered screw ship. But instead of hamsters I would use what ever animal is the lightest and fastest available and preferably only found on the island. Such as ostriches or some other animal that excels at high speed or long endurance running.
An interesting plot point could be if the best runners to power the ships only existed on the island. This would provide them with a sight advantage if they could maintain exclusive control of this resource. There could be some neat espionage stuff too.
Otherwise things will tend to reach parity.
One Idea I did have the concerns magic, is if you allow enchantments. Then magic can be used that does not involve active participation of the spell-caster. What form of enchantments could vary from some kind of "magical" engine, to harnesses to control sea beasts. Even a jar of wind or something like that ( something similar to this was used in Homers Odyssey )
Odysseus and his crew stayed with Aeolus, a king endowed by the gods with the winds. He gave Odysseus a leather bag containing all the winds, except the west wind, a gift that should have ensured a safe return home. Just as Ithaca came into sight, the greedy sailors naively opened the bag while Odysseus slept, thinking it contained gold. All of the winds flew out and the resulting storm drove the ships back the way they had come.
That gives me another odd idea for a story, what if the magicians secretly enchanted the island and stole the winds, just so they could sell it back in small amounts to the populous ... lol