كتاب الأشربة
36
The Book of Drinks
(18)
Chapter: It is recommended to lick one's fingers and wipe the bowl, and to eat a piece of food that is dropped after removing any dirt on it. It is disliked to wipe one's hand before licking it, because of the possibility that the blessing of the food may be in that remaining part. The Sunnah is to eat with three fingers
(18)
باب اسْتِحْبَابِ لَعْقِ الأَصَابِعِ وَالْقَصْعَةِ وَأَكْلِ اللُّقْمَةِ السَّاقِطَةِ بَعْدَ مَسْحِ مَا يُصِيبُهَا مِنْ أَذًى وَكَرَاهَةِ مَسْحِ الْيَدِ قَبْلَ لَعْقِهَا
Sahih Muslim 2032 c

'Abdullah b. Ka'b reported that his father Ka'b narrated to him that Allah's Messenger (ﷺ) used to eat with three fingers and when he had finished (eating), he licked them.

وَحَدَّثَنَا مُحَمَّدُ بْنُ عَبْدِ اللَّهِ بْنِ نُمَيْرٍ، حَدَّثَنَا أَبِي، حَدَّثَنَا هِشَامٌ، عَنْ عَبْدِ الرَّحْمَنِ، بْنِ سَعْدٍ أَنَّ عَبْدَ الرَّحْمَنِ بْنَ كَعْبِ بْنِ مَالِكٍ، - أَوْ عَبْدَ اللَّهِ بْنَ كَعْبٍ - أَخْبَرَهُ عَنْ أَبِيهِ، كَعْبٍ أَنَّهُ حَدَّثَهُمْ أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم كَانَ يَأْكُلُ بِثَلاَثِ أَصَابِعَ فَإِذَا فَرَغَ لَعِقَهَا ‏.‏
Reference : Sahih Muslim 2032 c
In-book reference : Book 36, Hadith 172
USC-MSA web (English) reference : Book 23, Hadith 5041
  (deprecated numbering scheme)