That the Buddha existed is not seriously disputed. When he lived is, and the disagreement is large enough to matter for anything built on it.
Two chronologies compete. The long chronology, preserved in the Sri Lankan Pali sources, dates his death to 544 or 486 BCE depending on how the regnal reckoning is handled, and puts his life roughly at 563-483 BCE. That is the figure in most older textbooks. The short chronology, derived from Indian and Chinese sources, places the death around 368 BCE. Since the 1980s a scholarly reassessment, associated particularly with Heinz Bechert, has pushed the majority view toward a date of death somewhere between about 410 and 370 BCE, that is, roughly a century later than the traditional figure.
This is not a technicality. The dating is anchored to the reign of Ashoka, which is fixed independently through his inscriptions naming Hellenistic kings, and the two chronologies place different intervals between the Buddha and Ashoka. Moving the Buddha a century later compresses the time available for the early schisms and for the development of the canon.
Nothing he said was written down in his lifetime. The Pali canon was transmitted orally and, by its own tradition, committed to writing in Sri Lanka in the first century BCE. Oral transmission in this culture was formal, communal and heavily mnemonic rather than casual, and the texts show the marks of it in their repetition and formulae, which is evidence of care in transmission, not a guarantee of the words.
Archaeology at Lumbini has produced a timber structure under the later shrine, radiocarbon dated to the sixth century BCE, which its excavators read as supporting the earlier chronology. Others dispute what the structure was.