During the lifetime of the Prophet (salallahu alayhi wa sallam), there was no pressing need to write down all of his various statements or record his actions because he was present and could be consulted at any time. As a matter of fact, the Prophet (salallahu alayhi wa sallam) himself made a general prohibition against writing down his statements which were other than the Qur'an itself. This was to prevent the possibility of mixing up the Qur'an with his own words during the era of revelation. Consequently, the greatest stress regarding writing was placed on recording the Qur'anic verses.
However, there are many authentic narrations collected by the Scholars of Hadeeth, that prove that hadeeth were recorded in writing even during the lifetime of the Prophet (salallahu alayhi wa sallam). For example, 'Abdullah bn Amr (RA) said: "I used to write everything which I heard from the Messenger of Allah with the intention of memorizing it. However, some Qurayshites forbade me from doing so saying, 'Do you write everything that you hear from him, while the Messenger of Allah is a human who speaks in anger and pleasure?' So I stopped writing, and mentioned it to the Messenger of Allah. He pointed to his mouth and said,
"Write! By Him in whose hand is my soul, only truth comes out from it.'" (Abu Daud)
Abu Hurayrah (RA) said: 'When Makkah was conquered, the Prophet stood up and gave a sermon (Abu Hurayrah then mentioned the sermon). A man from Yemen, called Abu Shah got up and said, 'O Messenger of Allah! Write it down for me." The Messenger of Allaah replied: write it for Abu Shah." (Abu Daud) A1-Waleed asked Abu 'Amr, "what are the writing?" He replied, "The sermon which he heard that day." (Abu Daud)
Abu Qabeel said: 'We were with 'Abdullah ibn Amr bn Aas (RA) and he was asked which city will be conquered first Constantinople or Rome? So Abdullah called for a sealed trunk and he said, ''Take out the book from it." Then Abdullaah (RA) said, 'While we were with the Messenger of A1laah writing, the Messenger of Allah was asked, ‘Which city will be conquered first, Constantinople or Rome?" So Allah's Messenger (salallahu alayhi wa sallam) said, 'The city of Heraclius will be conquered.