Did the Messenger of Allah, may Allah bless him and grant him peace, fast nine days of Shawwal?

It has not been established that the Prophet, peace and blessings be upon him, fasted nine days of Shawwal. Rather, what has been established is his encouragement to fast six days of it, by his saying: "Whoever fasts Ramadan then follows it with six days of Shawwal, it is as if he has fasted the entire year." Perhaps the questioner intended the nine days of Dhul-Hijjah, for it has been established that he fasted them.