As Lewis Hamilton’s fame within the Formula 1 world has soared, one rather uncharacteristic thing about the seven-time champion has public persona has been his close bond with his pet dog Roscoe.

Bulldog Roscoe has been by the F1 star’s side since his career really took off—and has always stayed in first place.

Ferrari has said it still has faith in Hamilton, though F1’s former boss Bernie Ecclestone has urged him to call time on his ...