Are Muslims descendants of Ishmael the first son of Abraham?

Answer 1

No. There is no evidence that the Arabs believed they were descended from Ishmael prior to the advent of Islam. Islam taught the Arabs to accept as true much of what is in the Old Testament, so from this time forward, many Arabs have believed they really are descended from Ishmael.

Of course, the biblical story of Arabic descent from Ishmael ought to apply to all Arabs, Christian or Muslim. It does not apply to Muslims not of pure Arab descent.
 

Answer 2

First & as it's stated in the above answer the term Muslim is different from the term Arabian. The Majority of Muslims are non-Arabs & the Majority of Arabs are Muslims.

To answer your question as it is the answer is No, because you are asking about the Muslims not about the Arabs. Originally, Muslims are the followers of the religion of Islam so they are generally not the descendants of any specific person as they can be any people of any race.

Now, if the inquiry is about the Arabs whose majority are Muslims. Then the answer is that some of the Arabs (not all) are descended from Ishmael PBUH. Prophet Abraham brought Hajar (his Egyptian second wife) and Ishmael (or Ismael) while still child to a place close to the foundations of Kaba in Mecca and left them per God command to him. Then as Ishmael groew up he helped his father Abraham in raising the foundation of Kaba. In the Holy Quran, The Almighty God says:
(And remember Abraham and Isma'il raised the foundations of the House (With this prayer): "Our Lord! Accept (this service) from us: For Thou art the All-Hearing, the All-knowing.
128. "Our Lord! make of us Muslims, bowing to Thy (Will), and of our progeny a people Muslim, bowing to Thy (will); and show us our place for the celebration of (due) rites; and turn unto us (in Mercy); for Thou art the Oft-Returning, Most Merciful.
129. "Our Lord! send amongst them an Messenger of their own, who shall rehearse Thy Signs to them and instruct them in scripture and wisdom, and sanctify them: For Thou art the Exalted in Might, the Wise).
Holy Quran,(2 : 127-129)

The Messenger in the supplication that is mentioned in the last verse is Muhammad PBUH who is an Arabian prophet. Also,He PBUH Himself assured that Ishmael PBUH is His grandfather in many sayings of Him.

Ishmael marriedc an Arabian wife. In the Arabian historical books that studied in much details the Arabian major tribes (Adnan,Qahtan & Aad),their lives,origins,homelands & achievements. It's assured that Ishmael PBUH is the grandfather of the major Arabian tribe (Adnan) from which Muhammad PBUH is descended.
P.S. Most (not all) of the Arabian tribe (Aad) had perished due to the disbelief of their majority in the Arabian prophet Hud " spelt as who'd" PBUH (The prophet with the few believers in his message were saved from that punishment & consequently they had survived).  

Answer 3

Not all Muslims Arabs (but more than 75% of Muslims are non Arabs) and not all Arabs are prophet Ishmael (or Ismael) descendents.