Were all the verses of the Quran written down during the lifetime of the Prophet (s.a.w.)?

We know that during the lifetime of the Prophet (s.a.w.), scribes and other companions used to write down verses of the Quran. But by the time the Prophet (s.a.w.) died, did all the verses of the Quran exist in written form? I was told that during the lifetime of the Prophet (s.a.w.), not all verses were written down. Only some verses were written down while other verses were only memorized. Please provide sources/evidences for your answer.