The British abolished the Peshwaship at the time of Baji Rao II. The Peshwaship was a hereditary position of prime minister in the Maratha Empire. Option 1, Raghunath Rao, was a claimant to the Peshwiship, but he did not become the Peshwa. Option 2, Narayan Rao, was the Peshwa before his assassination, but the British did not abolish the Peshwaship during his time. Option 3, Madhav Rao II, succeeded Narayan Rao as the Peshwa, but the British also did not abolish the Peshwaship during his tenure. Therefore, the correct answer is option 4, Baji Rao II, as it was during his time that the British abolished the Peshwaship.