Source: Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Kitāb al-Ṣalāh, Vol. 1, p. 159
Clarified for Situational Etiquette in Worship
According to Maulana Thanaullah Amritsari, a respected scholar,
"There is no hadith that prohibits giving salām during the adhān."
(Refer to: Fatāwā Thanāʾiyyah, Vol. 1, p. 323)
However, it is preferable for a person to maintain reverence and focus during the adhān and respond to its words, as per the Sunnah.
Reference:
Fatāwā Thanāʾiyyah, Vol. 1, p. 323
Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Vol. 1, p. 159
Clarified for Situational Etiquette in Worship
❖ Question
Is it permissible to say salām to someone while the adhān (call to prayer) is being recited?❖ Answer
✔ There is no explicit prohibition in any authentic hadith or Sharʿī evidence that forbids saying salām during the adhān.
"There is no hadith that prohibits giving salām during the adhān."
(Refer to: Fatāwā Thanāʾiyyah, Vol. 1, p. 323)

Reference:
Fatāwā Thanāʾiyyah, Vol. 1, p. 323
Fatāwā ʿUlamāʾ Ahl al-Ḥadīth, Vol. 1, p. 159