Did the Prophet (peace be upon him) or the Salaf pray Tarawih all night, meaning from after 'Isha prayer until Fajr?

It was not narrated that the Prophet (peace be upon him) prayed Tarawih from after Isha until Fajr. Nor was it his guidance to pray the entire night; rather, he would sleep and then pray. He did not exceed eleven rak'ahs in Ramadan or other months. The best prayer is the prayer of Dawud (peace be upon him), who would sleep half the night, pray for a third of it, and then sleep for a sixth of it. It was reported that he (the Prophet) prayed on some nights until close to Fajr, especially during the last ten nights of Ramadan. On some nights, he would prolong the prayer so much that they almost missed the suhoor (pre-dawn meal).

As for what was reported from the Salaf (early Muslims), it was narrated that Umar ibn al-Khattab (may Allah be pleased with him) gathered the people to pray twenty-one rak'ahs, reciting hundreds of verses, and they would finish close to the dawn. It was also reported from Tamim al-Dari, Abdullah ibn Mas'ud, and some of the Salaf that they would pray the entire night or most of it. This is not considered a contradiction to the guidance of the Prophet (peace be upon him), as he did not specify a number of rak'ahs for Qiyam al-Layl (night prayer) and left the matter to the Muslim's ability, provided that no obligatory duty or right is neglected. Indeed, the Prophet (peace be upon him) forbade perpetual fasting and praying the entire night.

In summary: It was not explicitly proven that the Prophet (peace be upon him) prayed Tarawih from after Isha until Fajr, except in the hadith of Abu Dharr. However, this was indeed established through the actions of the Companions and the Tabi'in (generation after the Companions) during the time of Umar ibn al-Khattab (may Allah be pleased with him). There is no harm in this, while observing obligations and rights.