A residential university that opened before Europe had one
The great monastic university of Nalanda in northeast India was founded around 427 CE and drew thousands of students from China, Tibet, Korea, and beyond for its logic, medicine, and Buddhist philosophy. It had already been teaching for well over six centuries before Europe's oldest universities, Bologna and Oxford, opened their doors.
How we know
Nalanda founded c. 427 CE (Kumaragupta I / Shakraditya; corroborated by seal and numismatic evidence), flourished until its destruction c. 1200 CE (Bakhtiyar Khalji raid; Wikipedia Nalanda mahavihara gives "c. 1200 CE" as scholarly consensus — the popular date of 1193 CE is not the careful scholarly position). University of Bologna traditionally dated to 1088 CE (teaching began); Oxford teaching from c. 1096 CE. 1088 − 427 = 661-year head start. Sources: Wikipedia/Nalanda_mahavihara; Wikipedia/List_of_oldest_universities_in_continuous_operation; official Nalanda University history page (nalandauniv.edu.in).
One of many fact-checked wonders across eight traditions at The Spring of Wisdom.