So, we're looking for a plant-based material that'd be used to construct a floating seaborne island large enough for humans to live on it, which could then be self-sufficient, can be repaired and expanded, and which would remain far from land? In which case, I nominate Southern Bull kelp, aka as Durvillea Antarctica. Also known as "Rimurapa" by the Maori, "Rimuroa" by the Moriori, and as "Chocayuyo" by the Quechua, Aymara and Mapuche.
This species of brown seaweed, found across the Southern Ocean all the way to the shorelines of Antarctic, doesn't have air bladders- the seas of the Roaring Forties, Furious Fifties and the Shrieking Sixties are far too rough, raging and turbulent for that. Instead, it floats due to a unique honeycomb structure within the alga's blades, which also gives it exceptional strength and buoyancy, and helps the kelp avoid being damaged by storms. D. antarctica has to resist forces equivalent to winds of over 1100 km/h.
This is thought to be responsible for the wide circumpolar distribution of this genus, as the kelp is able to float when its holdfast fails (usually as the result of worms and molluscs eating away at it, or of the rocks it's latched onto breaking away). It can colonise other coastlines in this manner, and has been shown to carry communities of invertebrates across vast ocean distances from one shore to another, with this 'rafting' with Durvillaea antarctica especially allowing a wide range of species to island-hop and to re-colonise sub-Antarctic shores scoured clean by ice during the last Ice Age.
The southern Māori tribe Ngāi Tahu had various uses for bull kelp: the narrow stalk, connecting the holdfast to the blade, was fashioned into a flute; the blade was roasted and chewed; the holdfast stems or stipes were used to make bouncing balls; and wide blades were used as bags for preserving food. Māori made bags called pōhā by splitting open the blades and inflating them.
They produced the bags in large quantities during summer in preparation for the muttonbirding [titi] season. Inflated blades were hung up to dry for several days, then deflated and rolled up. These bags would then be used to store the mutton birds (titi) in their own fat. The outer skin of the blades is airtight, and traps air in the honeycomb-like structure inside each blade. Food preserved inside a pōhā can be kept safely for up to two to three years.
So, to create our massive, floating island style raft, we could either craft it by weaving the strands of kelp together- the simplest method, and the easiest to maintain and repair. Or (/and) we could employ the same method which the Maori used to produce their pōhā to split open and inflate every blade of bull kelp used in the raft's base, greatly increasing its buoyancy and making it a massive inflatable boat. Or, if you want something a bit more advanced and cutting edge, you could simply harvest all of that bull kelp (since it's a brown alginate), industrially process it to extract sodium alginate from it, and use them to mould the floating island vessel's hull out of alginate plastic instead.
Studies have already been conducted into using alginate polymers as building materials, and they've been shown to confer significant flame-, fire- and heat-resistance, along with imperviousness to water. Other substances can be added to the materials to improve cross-linking and/or to produce resistance to fungal degradation. The building materials which can be produced, processed, or treated using alginate polymers include, but are not limited to, wood products; masonry products; wall, roofing, flooring and siding products; and paint products. Furthermore, sodium alginate in the form of a gel may be used as a firebreak to effectively stop the advance of any fires which might endanger the island.
The agar and carrageenan which can be extracted from it are also very useful. As are the hydrocolloids extracted from this seaweed, which have molecules that give viscosity to creams and lotions. In fact, almost all cosmetics include among their ingredients “seaweed extract,” that’s why it is so common that they are incorporated in this class of products. Studies now show that this crop can also be readily used for the relatively cheap, simple and easy production of biofuels such as ethanol, which would doubtless be critical for establishing an industrial base, and sustaining the floating island-raft's propulsion. And to help on the sustainability front too, Cochayuyo is also fully edible; a much loved delicacy in Chilean cuisine.
Today it is appreciated for its remarkable medicinal properties. It is recommended for weight control, to lower cholesterol, to prevent constipation, goiter, hypothyroidism, heartburn, plus it is an energizer thanks to its vitamins and minerals. It can be eaten in multiple ways: as jam, “empanadas” , “ceviche”, grilled or mashed, croquettes, and even seaweed risottos. But its most common use is in stews like “charquicán”, a typical Chilean dish as well as in salads.
It's commonly preserved by being sun-dried, and harvests can be kept as stock for years- to prepare it in a dish, it simply needs to be softened up by being soaked in water. It also has some of the highest potential yields of any crop, thanks to its extremely fast growth rate, high calorie count, high protein content (enough to replace meat altogether in some native Chilean diets) and an extremely high mineral content (enough to also serve as an extremely useful green fertilizer- albeit also with a very high salt content- meaning that other salt-resistant crops could be grown as well on the island, even without soil).
So, just how nutritious is it? Here's a study of the Southern Bull Kelp's nutritional properties and maximum yields, taken in the waters of Kerguelen Islands:
The approximate composition (% dry weight) of Durvillaea antartica
(Cham.) is similar for the frond and stipe, being 28% ash, 3% protein,
ca 1% lipid, less than 10% acid-soluble carbohydrate, and 60%
acid-insoluble carbohydrate. The holdfast differed by being 22% ash
and 66% acid-insoluble carbohydrate. The energetic level of all 3
components was ca 13 kJ (/g dry wt). and ca 17 kJ (/g ash-free dry
wt). The relative proportion of the 3 plant components varied little
for plants having stipe diameters ranging from 10 to 40 mm, being ca
8, 3, and 89 % wet wt, and ca 10, 4, and 86 % kJ, for the holdfast,
stipe, and frond, respectively. Maximal density found in summer was
471 individual plants/sq m, 226 kg wet wt /sq m, and 457 000 kJ/sq m.
To put this in context- most modern countries have cereal yields equivalent to 5 tonnes per hectare. The world record yield for a rice harvest is 22.4 tonnes of rice per hectare; and the present day global production of raw plant calories from staple cereal crops is adequate to feed roughly 10 people/ha. In contrast, even when growing in the wild, largely uncultivated, Durvillaea antarctica in the waters of the Kerguelen Plateau delivers a maximum yield equivalent to 2,260 tonnes per ha, or 350 tonnes per ha in dry weight. In seaweed farms, yields can easily be twice that. And going by the nutritional values of the Durvillaea antarctica samples from the Kerguelen Plateau, that'd equate to a crop yield of roughly 2 million kJ per wet ton. The average adult individual's RDA is just under 1 million kJ per annum; thus, farming Cochayuyo for food, as well as for industrial purposes and for repairing and expanding the vessel, would allow them to feed one person for every single sq.m of cultivation area; 10000 people/ha.
So, just how large do you want your floating island to be, and what sort of population would you ideally want it to be capable of supporting? Durvillea antarctica grows at a depth of up to 15m below the surface; so, let's add sloped skirts around the 'coastline' of our island, going down to this depth, to provide an anchoring area for their holdfasts and serve as our cultivation area; roughly 30sqm for every meter of 'shoreline'.
If you do, an island the size of a large cruise ship would also produce a Cochayuyo harvest large enough to support a population of more than 20,000 people; an island the size of the Project Habbakuk Aircraft Carrier would be capable of growing at least enough Cochayuyo to feed roughly 84,000 people indefinitely. And the cultivation area could easily be increased five-fold or more by changing the shape of the island's 'coast', giving it a leaved or petaled shape, thereby increasing your maximum sustainable population until it's larger than Iceland's population.
Or, to present a less dystopian best-case scenario, to support a population of around 25,000 people in relative comfort, even with only 6% of the annual cochayuyo harvest used for food, and the vast majority of the harvest utilized for other purposes such as the creation of biofuels, biopolymers and bioplastics, skin products, cosmetics, and fertilisers to grow other crops in greenhouses on-board. How does that sound?