Source: Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Kitāb al-Ṣalāh, Vol. 1, p. 158
Clarified for Adab in Ṣalāh and Interaction with Adhān
✔ Since there is no authentic evidence permitting this within the prayer, the correct approach is to refrain from doing so.
Introducing an act into Ṣalāh that lacks Prophetic precedent is against the principles of Sunnah, and such additions should be avoided for maintaining the sanctity and structure of the prayer.
Reference:
Ahl Ḥadīth Suhādrah, Vol. 9, Issue 22
Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Vol. 1, p. 158
Clarified for Adab in Ṣalāh and Interaction with Adhān
❖ Question
If a person is praying and the Muʾadhdhin mentions the blessed name of the Prophet ﷺ during the Adhān, is it permissible for the one praying to say "Ṣallā Allāhu ʿalayhi wa sallam"?❖ Answer
✘ No, saying Ṣalawāt (durūd) during Ṣalāh upon hearing the Prophet’s ﷺ name is not legislated in the Sharīʿah.✔ Since there is no authentic evidence permitting this within the prayer, the correct approach is to refrain from doing so.

Reference:
Ahl Ḥadīth Suhādrah, Vol. 9, Issue 22
Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Vol. 1, p. 158