Alsalam Alikom, I would like to know if the Prophet (pbuh) said that" Salman is part of my family" because this is what is written in my kids' islamic study book. Jazakom Allah Khairn

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

The Prophetic narration which you mentioned in the question is classified by the scholars as Dha’eef (weak) and is reported by At-Tabaraani and Al-Haakim .

Allaah Knows best.