Short answer
No. Islam teaches that each person lives one life on earth, dies once, waits in barzakh (the life of the grave), and is raised in his or her own body on the Day of Resurrection for judgement. The belief that souls pass into new bodies or creatures (tanasukh) contradicts the Quran, and Sunni scholars counted it as disbelief.
Sunni creed affirms one earthly life, death, the life of the grave and the bodily resurrection. Belief in the transmigration of souls (tanasukh) denies the resurrection and the reckoning as the Quran describes them, so the scholars of creed ruled it outside Islam.
The four schools agree on this.
Explanation
The Quran describes a person at death asking: "My Lord, send me back that I might do righteousness in that which I left behind." The answer is: "No! It is only a word he is saying; and behind them is a barrier until the Day they are resurrected" (23:99-100). There is no return to this world in another form. The barrier (barzakh) holds every soul until the resurrection.
The Quran counts our stages: "you were lifeless and He brought you to life; then He will cause you to die, then He will bring you [back] to life, and then to Him you will be returned" (2:28). One life, one death, then the life of the Hereafter. When a man asked who could give life to crumbled bones, Allah answered: "He will give them life who produced them the first time" (36:79). The same person, in the same body, is raised and judged.
The Prophet ﷺ said that when a person dies, his place in Jannah or the Fire is shown to him morning and evening until the Day of Resurrection. The soul is waiting, aware, not starting a new life elsewhere. Reward and punishment are settled in the Hereafter, not through rebirth as a higher or lower being.
What about people who feel they have lived before, or recall "past lives"? Muslims do not take such feelings as evidence, since memory, imagination and suggestion can all produce them. Read what happens after you die?, barzakh and the afterlife and Islam and other religions.
The evidence
Quran, Surah Al-Muminoon 23:99-100
حَتَّىٰٓ إِذَا جَآءَ أَحَدَهُمُ ٱلْمَوْتُ قَالَ رَبِّ ٱرْجِعُونِ ﴿٩٩﴾ لَعَلِّىٓ أَعْمَلُ صَٰلِحًۭا فِيمَا تَرَكْتُ ۚ كَلَّآ ۚ إِنَّهَا كَلِمَةٌ هُوَ قَآئِلُهَا ۖ وَمِن وَرَآئِهِم بَرْزَخٌ إِلَىٰ يَوْمِ يُبْعَثُونَ ﴿١٠٠﴾
[For such is the state of the disbelievers], until, when death comes to one of them, he says, "My Lord, send me back That I might do righteousness in that which I left behind." No! It is only a word he is saying; and behind them is a barrier until the Day they are resurrected.
Shows there is no return to this world after death.
Quran, Surah Al-Baqara 2:28
كَيْفَ تَكْفُرُونَ بِٱللَّهِ وَكُنتُمْ أَمْوَٰتًۭا فَأَحْيَٰكُمْ ۖ ثُمَّ يُمِيتُكُمْ ثُمَّ يُحْيِيكُمْ ثُمَّ إِلَيْهِ تُرْجَعُونَ ﴿٢٨﴾
How can you disbelieve in Allah when you were lifeless and He brought you to life; then He will cause you to die, then He will bring you [back] to life, and then to Him you will be returned.
Counts one life and one death before the Hereafter.
Quran, Surah Yaseen 36:79
قُلْ يُحْيِيهَا ٱلَّذِىٓ أَنشَأَهَآ أَوَّلَ مَرَّةٍۢ ۖ وَهُوَ بِكُلِّ خَلْقٍ عَلِيمٌ ﴿٧٩﴾
Say, "He will give them life who produced them the first time; and He is, of all creation, Knowing."
Shows the same person is raised again.
Sahih al-Bukhari 1379
Sahih... إِنَّ أَحَدَكُمْ إِذَا مَاتَ عُرِضَ عَلَيْهِ مَقْعَدُهُ بِالْغَدَاةِ وَالْعَشِيِّ ...
"When one of you dies, his place is shown to him morning and evening."
Shows the soul waits in barzakh rather than entering a new life.
Where the ruling is recorded
- al-Aqida al-Tahawiyya, Section on the resurrection and the recompense. Affirms the raising of the bodies and the recompense on the Day of Resurrection.
- al-Aqa'id al-Nasafiyya with Sharh al-Taftazani, Section on the resurrection. States that the resurrection of the bodies is true and answers those who deny it.
- al-Shifa (Qadi Iyad), Part four, chapter on which statements are disbelief. Counts belief in the transmigration of souls into other bodies among statements that are disbelief.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. 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.