By Paul Ziobro 

Target Corp.'s website went down Sunday morning, overwhelmed by shoppers clamoring for a piece of a Lilly Pulitzer collection that was selling for a fraction of the price of the Palm Beach label's luxury clothes.

The surge in demand for the 250-item line showed the battered discounter can still summon up some of the "Tar-zhay" glitz that helped the retailer grow into a national chain, but it also underscored that Target still has a way to go before its website is robust enough to fight off rivals like Amazon.com Inc.

The episode also punctuated a new era of social-media marketing in which fads explode quickly, coveted out-of-stock items pop up at three times the price on eBay, and even frustrated tweets and downed websites can feed into the messaging.

"It's not a marketing ploy, but it is a marketing advantage," said Dean Crutchfield, an independent brand consultant. "This to me creates more demand."

Target took its website off line for about 20 minutes in the dark hours of Sunday morning as early shoppers scanned through Lilly Pulitzer bathing suits, shift dresses and jumpsuits before stores opened at 8 a.m.

At other times during the morning, Target paced visits to the site, leaving many shoppers unable to access it or a shopping app. The company also delayed the online launch by two hours.

"When the traffic got heavier, we made the website inaccessible," Target spokesman Joshua Thomas said.

Target's designer partnerships are less about selling product and more about buzz. Such intermittent one-offs with high-end designers, called capsule collections, are part of what helps the retailer maintain its cheap-chic appeal, which sets it apart from other discounters and helps draw more affluent shoppers along with the thrifty.

Keeping a slim inventory borrows from the playbook of luxury brands. A clogged website can annoy customers, but it can also help get across the message that the company has a hit on its hands, a point Target emphasized even as it said it was sorry.

"We realize there is an extreme amount of excitement around this collaboration, and we apologize for any disappointment this may have caused our guests," Mr. Thompson, the spokesman, said.

Target planned to have the sale up fully live at 3 a.m. Central Time, but delayed it until 5 a.m. because of high traffic, Mr. Thomas said. The number of visits was similar to Black Friday sales around Thanksgiving.

Three hours into the sale, the entire selection was nearly sold out online.

Stores also had heavy traffic and early lines. Employees at the Target in Watchung, N.J., said the store was completely sold out of Lilly Pulitzer gear within 30 minutes of opening. Many customers had started lining up at 6 a.m., they said.

The episode was a less serious echo of the launch of Target's Missoni collection in 2011, when the website crashed and couldn't be restored for several hours.

That snafu came shortly after Target retook control of its Web operations from former partner Amazon, and it highlighted shortcomings of the chain's online infrastructure.

Shortly afterward, Target shook up its online leadership. Executives have since touted improvements to the website to enable it to handle massive traffic, including sales around Black Friday that generated record online results for the site.

Capsule collections have spread to chains including J.C. Penney, Gap and Kohl's, but they have been a mixed bag.

Target has been building buzz for the Palm Beach-inspired Lilly Pulitzer line, setting up a temporary shop in New York's Bryant Park and running a juice bar, offering free manicures and providing ping-pong tables.

The company also promoted the line in television ads that started last week. Based on how quickly the collection is selling, Mr. Thomas said much of Target's paid advertising will end Sunday, including banner ads for the Web. TV ads are likely to end Monday morning.

Write to Paul Ziobro at Paul.Ziobro@wsj.com

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