Is it possible for a PC to create a permanent floating castle or island?

Question

Assume a high level / epic campaign, and a PC who wanted to create a floating castle or a floating island. Are there any resources in Dungeons & Dragons 5th edition that can allow a player character to do this?

Answer 1

Yes

A few likely options come to mind:

• Wish. Direct and effective.
• Divine Intervention. Either the spontaneous will of a deity (possibly in return for some favor), or the Cleric ability of the same name.
• The rules allow for custom-researched spells. It's the DM's call what the research and casting involves, but this seems reasonable to me as a 9th level spell. A permanent version of the seventh level Reverse Gravity, third level Levitate, or a reshaped, permanent version of seventh level Forcecage.

As do a few unlikely or DM dependent options:

• A hilarious number of Immovable Rods and sufficiently large stones to form a foundation
• An equally ridiculous number of flying carpets/brooms
• A truly absurd number of enslaved befriended Air Elementals or your choice of flying, ethereal creatures
• @TimGrant recommends getting a Cloud Giant to give/loan/rent you a Solid Cloud (MM 150)
• In a similar vein, the Elemental Plane of Air also seems to have solid clouds, as well as floating rocks. These rocks may or may not still float if transported to the Material Plane
• The plane Acheron is made of giant floating iron cubes. Steal one or two of those and hope they still float in the Material Plane
• Ask your DM if the Dragon Turtle, being a quadruped, is sufficiently horselike for the Horseshoes of a Zephyr to allow it to float 4 inches off the ground. Sadly, the Tarrasque is bipedal
• The rules for creating a sentient magic item are, ah, nonspecific as to its size, so perhaps you could create a single enormous, sentient flying carpet

Answer 2

There's an epic spell from 3.5e that you could use as a template when researching your own 5e epic spell:

Raise Island

Conjuration (Creation)
Spellcraft DC: 38
Components: V, S, XP, Ritual
Casting Time: 65 days, 11 minutes
Range: 0 ft.
Area: 100-ft.-radius hemispherical island
Duration: Permanent
Saving Throw: None
Spell Resistance: No
To Develop: 360,000 gp; 8 days; 14,400 XP. Seed: conjure (DC 21). Factors: change area to 10-ft. radius, 30-ft. high cylinder (+2 DC), change radius to 100 ft. (+40 DC), change height to 1,000 feet (+133 DC), permanent (×5 DC). Mitigating factors: increase casting time by 10 minutes (-20 DC), increase casting time by 65 days (-130 DC), nineteen additional casters contributing epic spell slots (-361 DC), one additional caster contributing one 6th-level spell slot (-11 DC), burn 2,000 XP per epic caster (-400 DC), spell only works on liquid (ad hoc -20 DC).
The character can literally raise a new island from out of the sea, bringing to the surface a sandy or rocky but otherwise barren protrusion that is solid, stable, and permanently established. The island is roughly circular and about 200 feet in diameter. Raise island only works if the ocean is less than 1,000 feet deep where the spell is cast.

XP Cost

2,000 XP.

If you want it to float elsewhere, there's a 9th level spell for creating one's own personal demiplane including whatever climate one desires, etc.