Run a mile
Distance oneself from physically or, more often, emotionally. The phrases is used in circumstances where a person has made an advance without an expectation of a response, but, when a response is forthcoming, is shown to be unprepared for it and immediately retreats.
'Run a mile' appears to be an American expression and dates from the early 20th century. The first example of it that I can find in print is from The Modesto News-Herald, December 1927:
"The romantic figure which American movies have presented of the 'sheik,' as interpreted by the late Valentino, is far from accurate. The real 'sheik' is often a wrangling old fellow, from whom the American woman would run a mile"
The 'mile' isn't especially significant but just denotes an arbitrarily long distance, that is, 'run a long way away from'.
There is a proverb which was described as 'old' in 1860 - 'Falsehood would run a mile while Truth is putting on his boots'. This doesn't appear to be related to the 'run a mile' idiom though, which alludes to running away.
The implied speed involved when one 'runs a mile' may derive from the earlier phrase 'run a mile a minute', which alludes to running at an impossibly fast pace. This came into vogue, again in the USA, when the railways were developed in the 19th century and people were able to travel at a mile a minute for the first time.
'Run a mile' may also owe something to the earlier phrase 'run amok'. Although the meanings of the phrases aren't the same the alliteration may have encouraged the choice of 'mile' rather than some other distance.
See other phrases that were coined in the USA.