Did Prophet Muhammad (SAW) say that none of us infallibles will die a natural death? If so, does that imply that the Prophet (SAW) himself was also martyred?
Yes, there are many authentic Hadeeths from the Prophet Muhammad (SAWA) and Ahlul Bayt (AS) stating that all of the infallible will be either killed or poisoned (ما منا إلا مقتول أو مسموم).
This Hadeeth was narrated by Imam Hasan (AS) from his grand father the Prophet (SAWA) after the martyrdom of Ameerul Mo'mineen ( AS). (Kifayatul Athar, by al-Khazzaz , page 160-162).
The Prophet (SAWA) himself was also poisoned. There are plenty of authentic narrations in Sunni and Shia books stating that the Prophet did not have natural death but was poisoned. ( Al- Mustadrak by Al-Haakim al-Nisaboori, V.3, P.60) and (al-Seerah al-Nabawiyyah by Ibn Katheer , V. 4, P. 449) and (al-Tabaqat al- Kubra by Ibn Sa'ad , V. 2, P. 249). These are few of Sunni books.
Wassalam.