This should unlock naturally playing through official mode and a bit of story mode, but if you are missing any you can play the mission mode friendship mission "Good luck, Loser!" as there is always a random friendly ship on that mission.
The 10 Battleships are:
White Base
Salamis
Zanzibar
Musai
Elmeth
Argama
Radish
Gwadan
Ra Cailum
Rawloola