Is there a noble Hadith in which the Prophet, peace and blessings be upon him, said about Salman al-Farsi during the Battle of the Trench: "Towards the end of time, my Ummah will be victorious through the people of this (and he placed his hand on Salman al-Farsi's back)"?

There is no hadith with the stated meaning. However, it is narrated in Sahih Muslim that the Prophet, peace and blessings be upon him, said: "Even if faith were at the Pleiades, men from Persia would attain it."

Al-Tirmidhi, Ibn Hibban, and others narrated from Abu Hurairah, may Allah be pleased with him, that some companions of the Messenger of Allah, peace and blessings be upon him, said: "O Messenger of Allah, who are these people whom Allah mentioned that if we turn away, He will replace us with them, and they will not be like us?" The narrator said: "Salman was beside the Messenger of Allah, peace and blessings be upon him." So the Messenger of Allah, peace and blessings be upon him, struck Salman's thigh and said: "This man and his companions. By Him in Whose Hand is my soul, even if faith were tied to the Pleiades, men from Persia would attain it." This hadith was authenticated by Al-Albani.

Al-Mulla Ali Al-Qari said in Mirqat al-Mafatih: "'Men from Persia' means the Persian group generally, or those whose language is Persian, or those from the land of Persia."