Messenger of Allah ﷺ used to practice Iʿtikaf in the last ten days of the month of Ramadan.  

كَانَ رَسُولُ اللَّهِ ﷺ يَعْتَكِفُ الْعَشْرَ الأَوَاخِرَ مِنْ رَمَضَانَ  

(the wife of the Prophet) The Prophet ﷺ used to practice Iʿtikaf in the last ten days of Ramadan till he died and then his wives used to practice Iʿtikaf after him.  

زَوْجِ النَّبِيِّ ﷺ أَنَّ النَّبِيَّ ﷺ كَانَ يَعْتَكِفُ الْعَشْرَ الأَوَاخِرَ مِنْ رَمَضَانَ حَتَّى تَوَفَّاهُ اللَّهُ ثُمَّ اعْتَكَفَ أَزْوَاجُهُ مِنْ بَعْدِهِ  

Aisha said, "the Prophet ﷺ used to practice Iʿtikaf in the last ten days of Ramadan and I used to pitch a tent for him, and after offering the morning prayer, he used to enter the tent." Hafsa asked the permission of ʿAisha to pitch a tent for her and she allowed her and she pitched her tent. When Zainab bint Jahsh saw it, she pitched another tent. In the morning the Prophet ﷺ noticed the tents. He said, 'What is this?" He was told of the whole situation. Then the Prophet ﷺ said, "Do you think that they intended to do righteousness by doing this?" He therefore abandoned the Iʿtikaf in that month and practiced Iʿtikaf for ten days in the month of Shawwal."  

كَانَ النَّبِيُّ ﷺ يَعْتَكِفُ فِي الْعَشْرِ الأَوَاخِرِ مِنْ رَمَضَانَ فَكُنْتُ أَضْرِبُ لَهُ خِبَاءً فَيُصَلِّي الصُّبْحَ ثُمَّ يَدْخُلُهُ فَاسْتَأْذَنَتْ حَفْصَةُ عَائِشَةَ أَنْ تَضْرِبَ خِبَاءً فَأَذِنَتْ لَهَا فَضَرَبَتْ خِبَاءً فَلَمَّا رَأَتْهُ زَيْنَبُ ابْنَةُ جَحْشٍ ضَرَبَتْ خِبَاءً آخَرَ فَلَمَّا أَصْبَحَ النَّبِيُّ ﷺ رَأَى الأَخْبِيَةَ فَقَالَ مَا هَذَا فَأُخْبِرَ فَقَالَ النَّبِيُّ ﷺ آلْبِرُّ تُرَوْنَ بِهِنَّ فَتَرَكَ الاِعْتِكَافَ ذَلِكَ الشَّهْرَ ثُمَّ اعْتَكَفَ عَشْرًا مِنْ شَوَّالٍ  

With the start of the last ten days of Ramadan, the Prophet ﷺ used to tighten his waist belt (i.e. work hard) and used to pray all the night, and used to keep his family awake for the prayers.  

كَانَ النَّبِيُّ ﷺ إِذَا دَخَلَ الْعَشْرُ شَدَّ مِئْزَرَهُ وَأَحْيَا لَيْلَهُ وَأَيْقَظَ أَهْلَهُ