Short answer
Yes. The Prophet ﷺ wore a turban (imamah), so wearing one is a sunnah of his habits: rewarded when you wear it to follow him, but not obligatory and not sinful to leave. The Hanafi texts recommend letting its tail hang between the shoulders. The four schools agree.
The turban is a sunnah of habit (sunnah al-zawa'id), not of worship. Wearing it out of love for the Prophet ﷺ is rewarded, leaving it is not blameworthy, and letting a tail (adhabah) hang down between the shoulders is mustahabb (recommended).
The four schools agree on this.
Explanation
Jabir reported that the Prophet ﷺ entered Makkah on the day of its conquest wearing a black turban, and Amr ibn Hurayth saw him give a sermon in a black turban with its two ends hanging between his shoulders. The Companions and the scholars after them wore turbans, and in many Muslim lands it became the dress of scholars and elders.
What kind of sunnah. Hanafi scholars divide the Sunnah into acts of worship the Prophet did regularly, like the sunnah prayers, and his everyday habits in dress, food and manners. The turban belongs to the second kind. You are rewarded if you wear it to follow him, but a man who does not wear one is not falling short in his religion. The four schools agree that it is not obligatory.
Practical details. No authentic hadith fixes a length or a colour. Black and white are both reported, and the Hanafi texts recommend letting a tail hang down the back. You may wear it over a cap. In wudu you must wipe at least a quarter of the head itself, so loosen or lift the turban to do that; wiping only over the turban is not enough in the Hanafi school. See how to make wudu.
Weak reports. Narrations claiming that one prayer in a turban equals twenty-five or seventy prayers without one are not authentic, so do not pass them on. The reward is in following the Prophet with love and a good intention. For caps, see do Muslim men have to cover their heads, and for the rest of his dress, what the Prophet wore.
The evidence
Quran, Surah Al-Ahzaab 33:21
لَّقَدْ كَانَ لَكُمْ فِى رَسُولِ ٱللَّهِ أُسْوَةٌ حَسَنَةٌۭ لِّمَن كَانَ يَرْجُوا۟ ٱللَّهَ وَٱلْيَوْمَ ٱلْءَاخِرَ وَذَكَرَ ٱللَّهَ كَثِيرًۭا ﴿٢١﴾
There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often.
The Prophet is the model to follow, including in his manner of dress.
Quran, Surah Al-A'raaf 7:31
۞ يَٰبَنِىٓ ءَادَمَ خُذُوا۟ زِينَتَكُمْ عِندَ كُلِّ مَسْجِدٍۢ وَكُلُوا۟ وَٱشْرَبُوا۟ وَلَا تُسْرِفُوٓا۟ ۚ إِنَّهُۥ لَا يُحِبُّ ٱلْمُسْرِفِينَ ﴿٣١﴾
O children of Adam, take your adornment at every masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess.
Believers are told to dress well, especially for prayer.
Sahih Muslim 1358
Sahih... دخل يوم فتح مكة وعليه عمامة سوداء ...
"He entered Makkah on the day of the conquest wearing a black turban."
Shows that the Prophet wore a turban.
Sahih Muslim 1359
Sahih... وعليه عمامة سوداء قد أرخى طرفيها بين كتفيه ...
"He was wearing a black turban, and he had let its two ends hang down between his shoulders."
The basis for letting the tail of the turban hang down the back.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Taharah, sunnahs of wudu. Explains the two kinds of sunnah: sunnah of guidance and sunnah of habit, such as the Prophet's dress.
- al-Durr al-Mukhtar with Radd al-Muhtar, Kitab al-Hazr wa al-Ibaha, section on clothing. Recommends letting the tail of the turban hang between the shoulders.
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.