No. Just 'I'm late' or 'I'm running late'. Any promised time is presumed.
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Mister MicawberAny promised time is presumed.I see.
TakaBut this does work, right?I'm getting late for an appointment.No, you are running late (for an appointment).
TakaIs it that 'late for the time one promised' doesn't sound like English?It does not sound like natural English.