Is it permissible for a person to say that a revered companion of the Prophet is his ancestor, based on what is known and inherited within his tribe, and based on a family tree compiled by researchers from the tribe?

There is no objection to stating one's lineage to a Companion if the lineage is well-known and widespread, for people are trusted concerning their lineages. Lineage is established by circumstantial evidence such as a family tree, the testimony of just witnesses, or widespread fame and hearsay among people. However, one must beware of boasting about this lineage, as it is forbidden and reprehensible in Islam, and it may be considered arrogance, due to the Prophet's (peace and blessings be upon him) saying: "Indeed, Allah, the Mighty and Majestic, has removed from you the arrogance of the pre-Islamic era (Jahiliyyah) and its boasting about ancestors. [There is only] a pious believer and a wretched sinner. You are the children of Adam, and Adam was created from dust," and his saying: "Four things from the affairs of the pre-Islamic era (Jahiliyyah) will remain in my nation and they will not abandon them: boasting about ancestry, slandering lineages, seeking rain through the stars, and wailing."