Is it reported from the Companions that they forbade speaking to people after the dawn prayer?

There is no [textual] evidence that prohibits speaking after the Fajr prayer; rather, there is evidence that indicates its permissibility. The Prophet, peace and blessings be upon him, used to converse with his companions after the Fajr prayer. This implies the permissibility of speaking after the prayer and before sunrise, even if it's not dhikr (remembrance of Allah).

However, engaging in dhikr at this time is better, due to Allah's saying: {And glorify [Allah] with praise of your Lord before the rising of the sun and before its setting} [Taha: 130], and due to the hadith (definition): "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of a complete, complete, complete Hajj and 'Umrah (definition)."

Some of the Salaf, such as Ibn Mas'ud, Sa'id ibn Jubayr, and Ibrahim al-Nakha'i, disliked speaking after the two rak'ahs of Fajr until the Fajr prayer itself, except for the remembrance of Allah or what was absolutely necessary. However, some did not hold this view, among them 'Aishah, may Allah be pleased with her, as the Prophet, peace and blessings be upon him, used to speak with her after the two rak'ahs of Fajr if he had a need.