So I'm going to attack this problem at points 2,3, and 5 because the choice of building material isn’t the most problematic part. The biggest question is, how big will this thing have to be to support just one person? From there, we can scale. Let's figure out how big an island would have to be to support a single person.
The island/ecology
Now that we've got an estimate for how much we need to keep afloat, we can calculate the size of our 1-person island. Although other commenters have pointed out the need for compartmentalization and structural reinforcement, we're going for a minimum size, so I designed a spherical shell-like boat made of pozzolan. We're adding weight with this shell, but it's also displacing water- which is what we want. The general equation we're solving is $weight\ of\ shell + weight\ of\ soil = weight\ of\ water\ displaced$
Weight of shell
Weight of the shell is complicated, but I assumed a 1m thick wall which simplifies the volume to $4\pi r^2t*\frac{1500kg}{m^3} \approx 9000*r^2kg$
This is the maximum buoyancy we can have- others have proposed modular systems and the like, but this was just a thought experiment to see what's theoretically possible. Anything modular or reinforced will quite possibly never float at all.
Weight of soil
The heaviest thing actually on the boat will definitely be the soil needed to farm. We can avoid the weight of the freshwater ponds by storing it in containers suspended over the side of the ship- that way they'll actually help a little with buoyancy. A person needs ~2000 calories per day. On a boat in the middle of the ocean, we have the advantage of being able to harvest seaweed- in this case, it'd be Sargassum. Not very tasty, but edible, and certainly enough of it. I'll say this knocks our calorie requirement down to ~1000 calories per day, because we need to get other nutrients that seaweed doesn't have. The two plants that come to mind are potatoes and nuts, which grow well together. Unfortunately, these plants weigh a lot because of the soil they require- potatoes require about 10 liters of soil and peanuts about 16. Given the calories in a potato, how many potatoes a plant produces a year, and the soil required by one plant, I end up with 7,000 liters of soil per person per year. For peanuts, by a similar calculation, I end up with 5,000 liters of soil per person per year. With a soil density of 1.5kg/L, that's 18,000kg of soil we need to support and keep afloat. At minimum. For one person.
Calculations
Now we can return to our original question- substituting the equation earlier with our new numbers becomes $9000r^2kg+18000kg=\frac{4}{3}\pi r^3*1000\frac{kg}{m^3}$ This solves nicely to output 1500m as the radius for our spherical boat topped with soil. Here, I almost gave up on the question- there's no way a pre-modern society would be able to manufacture this, and it'd just be for one person. It does scale back with more people because the soil requirement grows linearly and the volume grows as the cube of the radius, but it's not going to get smaller. However, given that you're okay with a mile-long boat I'll push on.
Ocean currents
You say you've plotted a course that would allow you to float indefinitely in circles, and I'll admit I'm skeptical. I assume you're using the North Atlantic Gyre to circulate, but objects that are suspended in the gyre don't circulate nicely indefinitely- they're deflected into the middle of the gyre by the Coriolis force, which is why we have things like the Great Pacific Garbage Patch and the Sargasso Sea. Check out this video from the UCLA spin lab- it does a great job of explaining it: https://www.youtube.com/watch?v=yP6eG9iXmKc.
However, we've got a boat- a rudder wouldn't work because we don't have a keel, but maybe some sails on the surface could solve that. They'd be only raised when a favorable wind is blowing away from the center of the gyre, and that might be enough to keep you out of it.
The biggest problem with relying on ocean currents is that they are slow. It takes about 5 years for water to circulate around a gyre, and that'd be highly variable. This means that for time spans of multiple years at a time, you'd be in the middle of the ocean and essentially impossible to reach, let alone find. The Canary Current is especially slow- moving only 0.03 m/s means it could take you a couple years just to float from north to south. Even worse, this slow speed means you'd spend a year floating in freezing cold waters near the Arctic, then two years floating near the equator in the dreaded doldrums. That's why the biology would be vital- but your crops would die every time you moved north or south. Perhaps crop rotation could help? If you had two different strains of potatoes or nuts, you could swap them out as you got warmer or colder.
TL;DR:
I hope that gives you some perspective on the challenges you're facing. In essence, your floating island would have to be massive, you'd have to cultivate multiple types of crops and manage them perfectly, and you'd have to periodically avoid getting swept into the gyre. Trade would be nearly impossible- nobody will be able to find you, and as soon as a sailing ship leaves to trade with the mainland it'd be impossible to find the way back.
Of course, that assumes everything goes perfectly- but a lot of the other answers in this chain deal with problems like patching the ship or storms. However, those kind of problems are often the ideal focus points for stories such as yours. Good luck!