It seemed Plíšková was on course to reach the semifinals of the grass-court event that acts as a warmup for Wimbledon but surrendered her advantage against 20-year-old Belarusian Aryna Sabalenka to lose 6-3, 2-6, 7-6 (7/5).
Sabalenka will now meet 2008 champion Agnieszka Radwanska in the last four.
The Pole, continuing her return from a back injury, defeated Jelena Ostapenko 6-2, 7-5.
Meanwhile, top seed Caroline Wozniacki, the 2009 champion and last year’s beaten finalist, will face Angelique Kerber in Friday’s other semi-final.
World number two Wozniacki, the Australian Open champion, beat eighth seed Ashleigh Barty 6-4 6-3, while Kerber had a much tougher time before seeing off Russia’s Daria Kasatkina 6-1, 6-7 (3/7), 7-6 (7/3).
In the men’s event, British number one Kyle Edmund was unable to build on his win over compatriot Andy Murray.
Despite winning the first set, Edmund lost 5-7, 6-3, 6-1 to world number 90 Mikhail Kukushkin.
Two-time Wimbledon champion Murray is returning from a longstanding hip injury, with the Scot set to announce on Friday if he will compete in this year’s edition.