Does anyone know what was the first computer to have a
built-in real-time
clock?
Depends a bit on what you mean by "real time clock".
In earlier days the (salesman's) definition would include a line frequency
interrupt.
Under that definition most, if not all, 3rd-generation mainframes had one as
standard. For most pre-LSI minicomputers it was likely to be an option
rather than standard.
If you mean a clock that maintains time when power is off or gets the time
by radio then we are probably into the micro era.
Andy