TheWertle has translated information from the Kingdom Hearts II: Final Mix+ Ultimania Book about Extra Gummi Ship Missions, known as EX Missions, that appear in Kingdom Hearts II: Final Mix. To unlock them, you have to achieve at least S-Rank in a Mission Level before unlocking the EX Mission for that Mission Level. (For example, complete S-Rank in Asteroid Sweep Mission Level 2 to unlock Asteroid Sweep EX Mission 2.)

Asteroid Sweep
EX Mission 1 - Use a Ship with at least 50 Speed
EX Mission 2 - Use a Ship with at least 100 Power
EX Mission 3 - Use a Ship with at least 250 Power

Stardust Sweep
EX Mission 1 - Use at least 12 Wing Gummi-Blocks
EX Mission 2 - Use only Shooting-type Weapons (i.e. the first category of guns)
EX Mission 3 - Use at least 20 Shooting-type Weapons

Phantom Storm
EX Mission 1 - Use a pre-made Highwind Ship
EX Mission 2 - Use a ship with a maximum HP of no more than 50
EX Mission 3 - The total maximum HP of Gummi Ship and Tiny Ship(s) no more than 20

Splash Island
EX Mission 1 - Use a Ship with the Ability “Change Type” set
EX Mission 2 - Use a pre-made Invincible Ship
EX Mission 3 - Use a Ship without Weapons

Ancient Highway
EX Mission 1 - Use a Ship with at least 8 Wheel Gummi-Blocks
EX Mission 2 - Use a Ship with at least 8 Abilities set
EX Mission 3 - The total Speed of Gummi Ship and Tiny Ship(s) must be at least 80

Floating Island
EX Mission 1 - Use a “Special Model” Gummi Ship
EX Mission 2 - Use a Ship with at least 4 Lock-On-type Weapons
EX Mission 3 - Don’t bring Tiny Ships

Broken Highway
EX Mission 1 - Use a Ship with at least 6 Shield Gummi-Blocks
EX Mission 2 - Use a Ship with at least 2 Slash-type Weapons
EX Mission 3 - Use a Ship with at least 4 Slash-type Weapons

Sunlight Storm
EX Mission 1 - Use a Tiny Ship with at least 4 Strike-type Gummi-Blocks
EX Mission 2 - Use a pre-made Falcon Ship
EX Mission 3 - Use a Ship with a cost no more than 400

Assault of the Dreadnought
EX Mission 1 - Use a Ship with at least 2 Ultima Weapons
EX Mission 2 - Use a Ship with a size that’s no more than 3×3×3
EX Mission 3 - Use a Ship with a Drain Gummi-Block

Also, for those of you who own Kingdom Hearts II: Final Mix+ and are playing through on Critical Mode, if you need help with the Lingering Spirit (Enigmatic Soldier, Terra), GameFAQs’ LordGarvar has created a Critical Mode Strategy Guide FAQ for defeating him if you’re interested: