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 a

Ibn Ka'b b. Malik reported on the authority of his father that he saw Allah's Messenger (ﷺ) licking his three fingers (after having finished the food). Ibn Hatim made no mention of" three". This hadith is also narrated through another chain of transmitters.

حَدَّثَنَا أَبُو بَكْرِ بْنُ أَبِي شَيْبَةَ، وَزُهَيْرُ بْنُ حَرْبٍ، وَمُحَمَّدُ بْنُ حَاتِمٍ، قَالُوا حَدَّثَنَا ابْنُ، مَهْدِيٍّ عَنْ سُفْيَانَ، عَنْ سَعْدِ بْنِ إِبْرَاهِيمَ، عَنِ ابْنِ كَعْبِ بْنِ مَالِكٍ، عَنْ أَبِيهِ، قَالَ رَأَيْتُ النَّبِيَّ صلى الله عليه وسلم يَلْعَقُ أَصَابِعَهُ الثَّلاَثَ مِنَ الطَّعَامِ ‏.‏ وَلَمْ يَذْكُرِ ابْنُ حَاتِمٍ الثَّلاَثَ ‏.‏ وَقَالَ ابْنُ أَبِي شَيْبَةَ فِي رِوَايَتِهِ عَنْ عَبْدِ الرَّحْمَنِ بْنِ كَعْبٍ عَنْ أَبِيهِ ‏.‏
Reference : Sahih Muslim 2032a
In-book reference : Book 36, Hadith 170
USC-MSA web (English) reference : Book 23, Hadith 5039
  (deprecated numbering scheme)