British number one Heather Watson was level at one-set all with France's 32nd-seed Caroline Garcia when play was suspended because of bad light.
Watson lost the first set 6-1 and asked for the match to be stopped at just after 21:00 BST after taking the second set 6-3. Play will resume on Tuesday.
Earlier, Johanna Konta was beaten 6-2 6-2 by former Wimbledon champion Maria Sharapova on Centre Court.
And Naomi Broady lost 7-6 (7-5) 6-3 to Colombia's Mariana Duque-Marino.
The British number three was unable to match the success of her younger brother Liam, who won his first-round match earlier in the day.
Watson recovers from disastrous start
Watson was hoping to become the first British woman through to round two but made a disastrous start on court 12.
The 23-year-old from Guernsey struggled badly on serve in the first set and was given a warning by the umpire after throwing her racquet when she was broken for the second time.
But Watson's belief returned at the start of the second set when she improved the quality and velocity of her delivery and also made inroads into Garcia's serve for the first time.
Helped by some vocal home support, she levelled the match before being asked if she wanted to continue and deciding it was too dark.
They will resume as the second match on court 12 on Tuesday to fight out the deciding set.
Konta outclassed by Sharapova
Konta, 24, has moved up to 126th in the world after beating two top-20 players at Eastbourne last week.
But fourth-seed Sharapova had too much quality and experience for the British number two on her Centre Court debut.
"The first match of Wimbledon is never easy, especially against a crowd favourite after a great week last week," Sharapova told BBC Sport.
"I wanted to focus on my game and I'm happy I did that."
Australia-born Konta was given a spot on the show court after her exploits at the Aegon International were followed by the draw against the Russian.
Konta earned two of the finest wins of her career in Eastbourne - where she now lives - beating Russian world number eight Ekaterina Makarova and 20th-ranked Garbine Muguruza of Spain.
While the Briton went into the match on the back of an impressive run, Sharapova was making her first appearance since losing in the French Open fourth round earlier this month.
She was not at her fluent best, producing seven double faults, while Konta set up five break points during the match.
But she only managed to take one of those opportunities before Sharapova claimed victory in one hour and 22 minutes.
Broady unable to celebrate family double
Home supporters flocked to see if 25-year-old Broady could follow in her brother Liam's footsteps - earlier he won a five-set thriller on his Grand Slam debut - and earn a surprise place in the next stage.
But the world number 200, ranked 101 places below the South American, made too many unforced errors to worry her rival, who won in an hour and 27 minutes.
The Stockport player did have a set point when leading 5-4 in the opener, but squandered that chance with a wayward forehand.
"I went into the match quite confident because I have had a good grass-court season overall and took some top players to close matches," Broady said afterwards.
"But it is the little breaks that I need. I didn't play my best tennis but I gave her a close match.
"It is quite tough for British wildcards because you are almost judged on this one tournament.
"Some people are negative when a wildcard loses in the first round but they don't watch us over the rest of the year. I'm proud of myself."