Muwatta Malik, Book 2, Hadith 47 (In-Book) ⓘ

Yahya related to me from Malik that he saw Rabia ibn Abd ar- Rahman vomit several times when he was in the mosque and he did not leave, nor did he do wudu before he prayed. Yahya said that Malik was asked whether a man who vomited food had to do wudu and he said, "He does not have to do wudu, but he should rinse the inside of his mouth and wash his mouth out."

وَحَدَّثَنِي عَنْ مَالِكٍ، أَنَّهُ رَأَى رَبِيعَةَ بْنَ أَبِي عَبْدِ الرَّحْمَنِ يَقْلِسُ مِرَارًا وَهُوَ فِي الْمَسْجِدِ فَلاَ يَنْصَرِفُ وَلاَ يَتَوَضَّأُ حَتَّى يُصَلِّيَ ‏.‏ قَالَ يَحْيَى وَسُئِلَ مَالِكٌ عَنْ رَجُلٍ قَلَسَ طَعَامًا هَلْ عَلَيْهِ وُضُوءٌ فَقَالَ لَيْسَ عَلَيْهِ وُضُوءٌ وَلْيَتَمَضْمَضْ مِنْ ذَلِكَ وَلْيَغْسِلْ فَاهُ ‏.‏