Aside from the 10th Doctor, there's Madrid M. N'Saita, written MADRID for the most part.
MADRID
A ship captain… well, a SMALL ship captain. She owns and flies a small craft, designed for speed and maneuverability. She is snarky and a slightly vulgar from being on her own in her ship for a so long. But she talks about big ideas in spite of that fact.
UNKNOWN: She is actually a TARDIS. For serious. (look at the letters of her name: MADRID M N'SAITA – I'M A DAMN TARDIS… she swears a lot)
Something happened to her before the fall of the Timelords that put her out of time and space. Coming back messed up her infrastructure and shorted her heart through her chameleon circuit, creating Madrid.
The rest of her disguised itself as MADRID's ship, one of unbelievable speed with an impossible engine. Easy enough to imitate, when you can travel through time.
The Ship:
Reactionless Drive: curvature of space time (big ball of wibbly wobbly…) allows it to work. Uses gyro (made out of one super-heavy mass & one other mass with weird properties when electrified, spinning around each other) to achieve quick maneuvers, & engages reactionless drive mode for superfast speeds.
Engaging the drive turns one mass into a sort of space time paddle. Its motion is sustained in our plane of existence by the counterweight, and doesn't suffer the same resistance from space time as a paddle in water, so it pulls the ship along without any effective resistance.
There's a few things that aren't quite right with the ship aside from an impossible engine. A storage cabinet somewhere is too big, that sort of thing.
MADRID perceives the ship as being quite old, made by an alien race. She had the good fortune to find it abandoned, barely running, but alive on some moon (why was she there?) and repaired the power source.
The space-time-effective matter is convincing, but no such material exists
Some conversation about how the Time Lords would know if it did, if anyone would.
MADRID insists that it works, and takes the Dr on a joyride at some point