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The world number five lost 6-4 7-6 (7-3) to 21st-ranked Rybakina, whose reward is a meeting with Coco Gauff's conqueror Anastasija Sevastova in the quarter-finals.
Third seed Bianca Andreescu of Canada also lost in straight sets, Anett Kontaveit of Estonia beating her 6-3 6-3.
And it was the same story for fourth seed Iga Swiatek, though she at least took a set as she was beaten by Daria Kasatkina.
It took an impressive rally from Russia's Kasatkina to bounce back from losing the first en route to a 4-6 6-0 6-1 success.
She will now face Jelena Ostapenko, who beat Ons Jabeur 5-7 6-4 6-3.
In fact, top seed Aryna Sabalenka was the only seed to avoid a surprise exit on Wednesday as she cruised through.
The Belarussian, ranked fourth in the world, beat Alison Riske 6-1 6-4 in just over an hour to set up a clash with Camila Giorgi.
Giorgi had earlier followed up a win over defending champion Karolina Pliskova in the last round by beating Shelby Rogers 6-3 4-6 6-2.
There was no such string of shocks at Wednesday's other WTA event, the Bad Homburg Open in Germany, though first and second seeds Petra Kvitova and Victoria Azarenka needed three sets to progress.
Kvitova lost a second-set tie-break as she beat Ann Li 6-2 6-7 (4-7) 6-1 to reach the quarters, while Azarenka defeated Alize Cornet 6-4 3-6 7-6 (9-7).
Angelique Kerber, the fourth seed, progressed more smoothly, earning a comfortable 6-0 6-2 win over Russia's Anna Blinkova.
And Nadia Podoroska saw off the challenge of Patricia Maria Tig, winning 7-6 (7-4) 2-6 6-4.