The earliest printed use of the now iconic term was in The Daily Mirror on November 2 1963. It described the mass hysteria that seemed to follow the fab four as they went around the world as women flung themselves at them, motorcades became miles long and the screaming during gigs was so loud that the band couldn't hear themselves think and went out of time with each other.
The Beatles' first film 'A Hard Day's Night' captures this pandemonium in all of its frantic glory. That phrase (and subsequent song title) was coined by Ringo after he was describing how hard the band had been working all day, only to look outside as he was speaking to realise that it was actually night-time.