Gold medal went to Australia's Chloe Esposito, France's Elodie Clouvel took silver and Poland's Oktawia Nowacka the bronze. Meanwhile, despite making up considerable ground in the final combined round, GB's French finished in sixth place, with Murray two places behind in eighth.
Britain had medalled in all four Olympics since a women’s individual event was introduced. Murray picked up silver in 2012.
Earlier in the day, the show jumping competition was something of a demolition derby, with a number of riders being unseated by their horses. Harry Potter was just one of many to show little wizardry on the course at the Deodoro Stadium.
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French had better luck, going into the showjumping round in 15th position; she jumped a clear round with Arnold Massangana to take the maximum 300 points.
She finished with 794 points, going into the final running and shooting contest later on Friday in ninth position. She will therefore start the final combined round 53 seconds behind the head of the leaderboard.
Compatriot Murray had a less successful course with Up Class Girl, knocking three fences down to concede 21 points in faults and take 279 points.
She had gone into the contest in 14th place and ended in 18th.
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