A TARDIS was given to the Fourth Doctor by the Time Lords as payment for his assistance in deterring the history of the Daleks (TV: Genesis of the Daleks) which he did by influencing events in the Adelphine Cluster so that a race of robots would rise up against the Daleks. He refused it, however.
This TARDIS worked with the Doctor's TARDIS key, moulding itself in a blue police box with it's functioning chameleon circuit. It was also attuned to his mind: when it was first sent to him and recognised he was in distress, it shaped itself into an escape pod, even shrinking the interior dimensions down to comfort him. He later shaped it into the golden yacht of the Tralsammavarians. (PROSE: A Device of Death)