Is it permissible for the greeting of peace to be: "As-salamu alaykum wa rahmatullahi wa barakatuh" with the addition of prayers upon the Prophet (peace be upon him)?

Acts of worship are built upon Tawqif (divine injunction); that is, adhering strictly to what is explicitly stated in the texts, without addition or alteration. An example of this is the Prophet Muhammad's (peace be upon him) refusal to change the wording "Your Prophet" to "Your Messenger" in the sleeping supplication, saying: "No, [say] 'Your Prophet whom You have sent.'"

Scholars have explained this by the necessity of adhering to the transmitted wordings exactly as they are, as the reward may be linked to them. Similarly, Ibn Umar (may Allah be pleased with both of them) disapproved of adding prayers upon the Prophet (peace be upon him) after sneezing, basing his stance on the Prophet's (peace be upon him) teaching to simply say Alhamdulillah (Praise be to Allah) without any addition.

The customary greeting of peace is what was transmitted from Adam (peace be upon him) and his progeny, which is "Assalamu Alaykum" (Peace be upon you). Therefore, adding prayers upon the Prophet (peace be upon him) when giving salutations is considered a blameworthy addition and an innovation (bid'ah), because the best guidance is the guidance of Muhammad (peace be upon him).