According to a hadith, is Hazrat Mirza Ghulam Ahmad (as), the Promised Messiah, a descendant of Salman al-Farsi?

No, no hadith says that. The Holy Prophet Muhammad (saw) spoke of people from Salman’s nation, not of his descendants.

One sometimes hears that, according to a hadith, the Promised Messiah is a descendant of Salman al-Farsi, the Persian companion of the Holy Prophet Muhammad (saw). But this is not said anywhere. Nor could it have been said, because Salman (ra) left no descendants. Anyone who claims this can easily be embarrassed, since others know this fact.

The background is a different one. When the verse above was revealed, the companions asked who these later ones were. The Holy Prophet (saw) placed his hand on Salman (ra) and said that even if faith were at the Pleiades, people from among these would reach it.

The word “these” does not refer to Salman as an individual. It refers to his whole nation, the Persians. The Holy Prophet (saw) was therefore speaking of someone who would arise from this people, not of a descendant of Salman. This is how Hazrat Mirza Ghulam Ahmad (as), the Promised Messiah, is understood in this context, as a man of Persian descent who revives faith at a time when it seems to have left the earth.

This shows the beauty of the language of the Holy Prophet (saw). His words are chosen precisely. Whoever pays close attention to them will not go wrong. Mistakes arise only when people quote him carelessly or attribute words to him that he did not say.