Hmmm, you've got problems. From the question text I'm assuming the island is at the halfway point and is ~1 mile across, so the bridge only has to be .5 miles long. I'm also assuming they don't have access to magic. If these assumptions are incorrect please clarify in the OP.
If they lack time and nearby resources they're screwed. If they have both:
One possibility is not building a "bridge" per-se, but rather extending a pole out to the island. Once they get one pole across it becomes easier to shuttle people along with, say, a basket and rope.
Now getting a half mile long pole across the ravine is no easy feat. If it's made of wood it's weight will make it bend down below the islands edge, so it will have to start at an upward angle or use some magic material that won't bend. It also can't be one piece, so these refugees will need to have the ability to make perfect connections between poles.
On second thought, a better idea: build a ballistia with a string attached to the bolt. Shoot the island. Now you have a rope extending across and you can either make a rope bridge or pull people back and forth in a basket. The first people to cross will have to crawl along the rope, so that will be dangerous, but also exciting if this is for a story.
Edit: So the longest range you can get on a ballista is ~800 meters. This isn't accounting for pulling along a bunch of rope capable of holding a man's weight. Don't wanna calculate that, so I'm just gonna halve it and say 400 meters (magic elven rope would of course help here). This is well short of .5 miles, so either the gap would have to be smaller, the ballista better, or they would have to build/find a higher place to fire from. For realism I would go for a combination of 1 and 3, but all would help.
Edit 2: Better Solution
Actually had another thought too. I'm assuming that if they can build a super long ranged ballista they can also build a boat capable of crossing the water to the other side. If they can, then send some guys to carry the end of a rope to the opposite cliffs (you'll need some good rock climbers). Then, pull the rope taunt so it's level with the floating island. Then shuffle someone along it to set up the pulley they'll use to cross.
This is probably a more feasible solution than the ballista, as the former would have its range vastly limited by the rope, and would be gambling that the bolt would be lodged strongly enough to hold both the rope and person.
Edit 2.5: Better Solution Maths
I'm guesstimating that the rope would weight 25 lb per 100 ft. This is mostly pulled out of my a$$, but rope capable of holding 100 lb on Amazon looks like it weights about 5lb per 100 ft. We need to hold a 200 lb man, plus its gotta be strong enough to hold together with just its own weight (not sure what the math is on calculating this strength), so maybe a thicker rope weighing 25 lb per 100 ft would be the way to go. Since you need 10,000 ft of rope, this brings the total weight to only 2,500 lb. This sounds heavy, but it could be a lot worse. They would need to set up some machinery on the other side to hold it taunt (and get the end up the cliff), but it's doable I'd say.
Or just go with the ballista.