Short answer
No. Shaytan, whose name was Iblis, is a jinn, not an angel. The Quran says plainly "He was of the jinn" (18:50). Angels are made of light and never disobey Allah, while jinn are made of fire and have free will, which is how Iblis was able to refuse Allah's command.
The relied-upon view in Sunni creed, held by the Hanafi theologians, is that Iblis is from the jinn and is the father of the jinn devils. Angels are protected from disobedience, so Islam has no idea of a "fallen angel".
The four schools agree on this.
Explanation
Many people come to this question from the Christian idea of Satan as a fallen angel. The Quran gives a different picture. When Allah told the angels to prostrate to Adam, they all did except Iblis, and the Quran explains why he could refuse: "He was of the jinn and departed from the command of his Lord."
Angels and jinn are two different creations. The Prophet ﷺ said the angels were created from light and the jinn from a smokeless flame of fire. Iblis himself said to Allah, "You created me from fire" (7:12). Angels do not disobey Allah in anything He commands (66:6). Jinn, like humans, choose, and some obey while others rebel.
Why does the command to the angels include him? Iblis was among the angels at that time, worshipping with them and so included in the command, though he was not one of them. A few early reports said he came from a group of angels, but most scholars hold to the clear wording of 18:50.
"Shaytan" is also used for any rebellious jinn or human who calls to evil. Iblis is the chief shaytan, and his followers among the jinn are his helpers. You protect yourself from them with dhikr, the morning and evening adhkar, ayat al-kursi and the last two surahs of the Quran. Read more in Angels in Islam and Jinn in Islam.
The evidence
Quran, Surah Al-Kahf 18:50
وَإِذْ قُلْنَا لِلْمَلَٰٓئِكَةِ ٱسْجُدُوا۟ لِءَادَمَ فَسَجَدُوٓا۟ إِلَّآ إِبْلِيسَ كَانَ مِنَ ٱلْجِنِّ فَفَسَقَ عَنْ أَمْرِ رَبِّهِۦٓ ۗ أَفَتَتَّخِذُونَهُۥ وَذُرِّيَّتَهُۥٓ أَوْلِيَآءَ مِن دُونِى وَهُمْ لَكُمْ عَدُوٌّۢ ۚ بِئْسَ لِلظَّٰلِمِينَ بَدَلًۭا ﴿٥٠﴾
And [mention] when We said to the angels, "Prostrate to Adam," and they prostrated, except for Iblees. He was of the jinn and departed from the command of his Lord. Then will you take him and his descendants as allies other than Me while they are enemies to you? Wretched it is for the wrongdoers as an exchange.
States directly that Iblis was a jinn.
Quran, Surah Al-A'raaf 7:12
قَالَ مَا مَنَعَكَ أَلَّا تَسْجُدَ إِذْ أَمَرْتُكَ ۖ قَالَ أَنَا۠ خَيْرٌۭ مِّنْهُ خَلَقْتَنِى مِن نَّارٍۢ وَخَلَقْتَهُۥ مِن طِينٍۢ ﴿١٢﴾
[Allah] said, "What prevented you from prostrating when I commanded you?" [Satan] said, "I am better than him. You created me from fire and created him from clay."
Iblis says he was made of fire, unlike angels made of light.
Quran, Surah At-Tahrim 66:6
يَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوا۟ قُوٓا۟ أَنفُسَكُمْ وَأَهْلِيكُمْ نَارًۭا وَقُودُهَا ٱلنَّاسُ وَٱلْحِجَارَةُ عَلَيْهَا مَلَٰٓئِكَةٌ غِلَاظٌۭ شِدَادٌۭ لَّا يَعْصُونَ ٱللَّهَ مَآ أَمَرَهُمْ وَيَفْعَلُونَ مَا يُؤْمَرُونَ ﴿٦﴾
O you who have believed, protect yourselves and your families from a Fire whose fuel is people and stones, over which are [appointed] angels, harsh and severe; they do not disobey Allah in what He commands them but do what they are commanded.
Angels never disobey, so the one who refused could not have been an angel.
Sahih Muslim 2996
Sahih... خُلِقَتِ الْمَلاَئِكَةُ مِنْ نُورٍ وَخُلِقَ الْجَانُّ مِنْ مَارِجٍ مِنْ نَارٍ ...
"The angels were created from light, the jinn were created from a smokeless flame of fire."
Shows angels and jinn are different creations.
Where the ruling is recorded
- Tafsir al-Qurtubi, Commentary on Surah al-Kahf, verse on Iblis being of the jinn. Sets out the views on whether Iblis was an angel and the evidence that he was a jinn.
- Akam al-Marjan fi Ahkam al-Jann (al-Shibli al-Hanafi), Chapter on Iblis and his origin. A Hanafi work on the jinn that discusses the origin of Iblis.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.