Has it been reported that the Prophet (peace be upon him) used to seek refuge [with Allah] in every rak'ah of the prayer?

It has not been established that the Prophet, peace and blessings be upon him, sought refuge with Allah in every rak'ah. The hadith reported concerning his seeking refuge, peace and blessings be upon him, in prayer was narrated by Ahmad and Tirmidhi from Abu Sa'id al-Khudri, that the Prophet, peace and blessings be upon him, when he stood for prayer, he would begin and then say: "I seek refuge with Allah, the All-Hearing, the All-Knowing, from the accursed Satan, from his madness, his pride, and his evil whisper." This clearly indicates that this seeking of refuge occurs in the first rak'ah of the prayer. Scholars have differed on the permissibility of seeking refuge in prayer after the first rak'ah. Many scholars have favored the view that seeking refuge in prayer is specific to the first rak'ah, because the recitation in prayer is considered a single recitation.