Tethered Aerial Release Developed In Style

The Tethered Aerial Release Developed In Style (abbreviated TARDIS), unlike its namesake, was a hot-air balloon owned by Jackson Lake while he was under the impression he was the Doctor. Though the Tenth Doctor was surprised and disappointed by it at first, he later found it to be nothing short of sheer brilliance that Jackson Lake already possessed.

When piloted by the Doctor, Lake's "TARDIS" was key in the destruction of the CyberKing in 1851. (TV: The Next Doctor)

Behind the Scenes
It was never explained if Lake believed the Doctor's TARDIS was lost or believed the Chameleon Circuit was stuck.