Alison Riske took a 5-3 lead against Anett Kontavei in the opening set in the only match which got underway on a gloomy day in the East Midlands.
Riske was unable to press home her advantage, as the heavens opened to ensure the American and Estonian Kontaveit must return on Saturday to finish their last-eight encounter.
The unseeded duo will be in action after fourth seed Monica Puig has belatedly taken on lucky loser Tamira Paszek, with top seed Karolina Pliskova starting the action on Centre Court against Ashleigh Barty.
Tara Moore and Zheng Saisai will contest the second quarter-final on Centre Court, with the four who advance to the semi-finals facing the prospect of a second match later in the day.