The Doctor often had one close to hand.
The Fifth Doctor was the incarnation who was the biggest fan of cricket, and therefore frequently had one on his person. This was in evidence in Amsterdam, when he emptied his pockets, and their contents included a cricket ball. (TV: Arc of Infinity)
Obviously, he often used the ball to actually play the game. But there were a number of notable occasions on which he used cricket balls in unusual ways. While drifting in space outside an Urbankan spacecraft, he propelled himself back to the the TARDIS by throwing a cricket ball against the hull of Monarch's ship, catching the ball as it bounced back, and letting the its momentum propel him back to the doors of the TARDIS. (TV: Four to Doomsday) Later, he saved Tegan Jovanka's life by throwing a cricket ball at the Raston Warrior Robot threatening her. The impact was just enough to throw the robot's aim so that the javelin it fired missed. (PROSE: The Eight Doctors) On yet another occasion, he switched the Eye of Akasha for a cricket ball and thus saved Tegan from the Judoon. (COMIC: The Forgotten) He also at least once used a cricket ball to hit a Sontaran's probic vent, thus rendering the warrior temporarily immobile. (PROSE: Lords of the Storm)
Even though the Fifth Doctor was the most overt cricketer, several incarnations of the Doctor were known to possess cricket balls. The Second Doctor once produced a cricket ball to allow Ben to knock an RTC unit out of Queen Aysha's hands. (PROSE: Invasion of the Cat-People) The Fourth Doctor gave Harry Sullivan a cricket ball to use in a failed attempt to deactivate the automatic guard system on Nerva Beacon. The cricket ball was destroyed. (TV: The Ark in Space) The Seventh Doctor used a biodata pod disguised as a cricket ball to store the biodata of his Time Lord self, and later that of his human persona. (PROSE: Human Nature) He also carried around a cricket ball, as was revealed when he once emptied his pockets (PROSE: Bad Therapy) and when Chang Lee retrieved his belongings from Walker General Hospital. (PROSE: Doctor Who - The Novel of the Film) The Eighth Doctor, too, carried a cricket ball on his person. (PROSE: The Dying Days) Although it wasn't strictly speaking his cricket ball, the Tenth Doctor — as John Smith — retained enough muscle memory of bowling to correctly throw a cricket ball to start a chain of events to save a mother and her baby from a falling piano. (TV: Human Nature) In his eleventh incarnation, the Doctor kept a cricket ball in the TARDIS drawing room (GAME: TARDIS) and underneath the TARDIS console. He also used a cricket ball to show and Amy, Rory and himself that he didn't have a proper body. This led to the hypothesis that he wanted to annoy them because they had proper bodies. (TV: Amy's Choice)
There were also times when cricket balls weren't quite as they appeared. A perversion of the Event Synthesizer caused a cricket ball in the 1980s to exchange places with a grenade at a game in Stockbridge. When it left the bowler's hand, it was a cricket ball, but by the time it reached the batsman, it was a grenade. (COMIC: The Tides of Time) Also, one of the cricket balls that the Doctor had hit at the Cranleigh Hall cricket ground turned out to be a segment of the Key to Time, which was retrieved by Ace. (COMIC: Time & Time Again)