Did Arianism survive as Islam? St. John of Damascus (676 - 749) writes on his Critique of Islam:


  From that time to the present a false prophet named Mohammed has appeared in their midst. This man, after having chanced upon the Old and New Testaments and likewise, it seems, having conversed with an Arian monk, devised his own heresy.


Could Arianism have influenced Islam? What are the similarities and differences between Arianism and Islam? Did Arianism survive to the current day as Arianism, with Islam being a branch-off based on Arian beliefs?
Like @curiousdannii pointed out in his comment, Arian influences do not equal an Arian viewpoint or verbatim parroting. In Qur'anic teaching, worship or veneration of anything besides God is strictly forbidden, or haram. This helps define what a Muslim (or a Christian or Jew or Sabean for that matter) can be and cannot be.

&quot;Say: 'O People of the Scripture! Come to an agreement between us and you: that we shall worship none but Allah, and that we shall ascribe no partner unto him, and that none of us shall take others for lords beside Allah.' And if they turn away, then say, 'bear witness that we are they who have surrendered unto him [Muslims].'&quot;
— Surat al-Imran 3:64

Here the authors sought to separate the wheat from the chaff in terms of who is and is not submissive to God's will; they understood that a polytheist couldn't care less about their al'Ilah, and they also knew that Catholics venerated saints (take others for lords)and placed Jesus at the right hand of God (ascribing a partner to God). Such a practice was unacceptable to them, and it was here that the line was drawn in the sand.
According to Arian schools of thought, worship of the Logos/the only-begotten Son was not rejected in favor of a strictly monotheistic faith. Arius' initial issue was not with worship of Jesus, but with semantics over what exactly the terms 'only-begotten' and 'one substance' implied. One faction believed that Christ was of one-substance, and that the Logos had not been created by the Father. Arius' followers instead posited that in eternity past, the Father created a lesser divinity through whom the creation would arise and would act as our reflection of the deity, meaning that the Logos was technically a created being.

&quot;But we say and believe and have taught, and do teach, that the Son is not unbegotten, nor in any way part of the unbegotten; and that he does not derive his subsistence from any matter; but that by his own will and counsel he has subsisted before time and ages as perfect as God, only-begotten and unchangeable, and that before he was begotten, or created, or purposed, or established, he was not.&quot;
— Arius of Alexandria's Letter to Eusebius of Nicomedia (circa AD 300)

Here you can see that Islamic perspectives on God are very much distinct from Arianism, perhaps more compatible with the Adoptionism that was popular in early Christianity or Paul Samosatene's theology.