Investigators scrambled to piece together a motive in a fatal shooting spree at a Washington mall Friday but would not rule out terrorism after a nearly 24-hour manhunt ended with a Turkish immigrant, who seemed to be in a “zombielike” state, surrendering to police.

Rep. Michael T. McCaul, Texas Republican and chairman of the House Homeland Security Committee, said on Fox News’ “Sunday Morning Futures” that law enforcement authorities were “playing it safe, saying there are no links to terrorism.”

“But I wouldn’t be surprised if we find out more details about this individual, as well as we did Mr. Rahami,” he said, referring to New York and New Jersey bombing suspect Ahmad Khan Rahami.

Arcan Cetin, 20, an immigrant from Turkey and legal permanent U.S. resident, was arrested Saturday night on five counts of first-degree murder.

A string of attacks, which also includes a stabbing at a mall in St. Cloud, Minnesota, that left nine people injured, could play a prominent role in the final weeks of the presidential race.

Indiana Gov. Mike Pence, who is Republican presidential nominee Donald Trump’s running mate, said the attacks show a clear need for immigration reform.

“We don’t know the motivation of the permanent legal resident in Washington state who took five American lives last night, but I think what you heard Donald Trump talking about was just simply we need common sense back in our immigration policies in this country,” Mr. Pence said on “Fox News Sunday.” “We need to begin to recognize that we’re at war with radical Islamic terrorism.

“Hillary Clinton won’t even name radical Islamic terrorism,” said Mr. Pence, referring to the Democratic presidential nominee. “Donald Trump has a plan to confront it. A part of that includes a full review of our immigration policies, suspending immigration from territories and nations that have been compromised by terrorism.”

Island County Sheriff’s Lt. Mike Hawley said he spotted Mr. Cetin from a patrol car in Oak Harbor, Washington, and immediately recognized him as the suspect in the deaths of five people at the Cascade Mall in nearby Burlington.

Lt. Hawley said at a news conference that police had received information that Mr. Cetin, of Oak Harbor, was in the area. Mr. Cetin was arrested before in the county on assault charges, Lt. Hawley said.

“I literally hit my brakes, did a quick turn, I jumped out,” Lt. Hawley said. “We both jumped out with our guns, and he just froze.”

Mr. Cetin was unarmed and was carrying a satchel with a computer inside. “He was kind of zombielike,” Lt. Hawley said.

Little is known about the suspect. Skagit County court records show three domestic violence charges against Mr. Cetin, reported by The Seattle Times, stemming from disputes with his stepfather.

He also was arrested on charges of drunken driving and told by an Island County District Court judge last year that he could not possess a firearm.

Investigators said they were searching Mr. Cetin’s apartment and car, and a court appearance was scheduled for Monday.

Social media accounts apparently belonging to Mr. Cetin show a fondness for the military and video games.

A Twitter account shows, among other things, various selfies, throwback photos, that he participated in paintball and that he “can’t wait for Halo 5,” a popular first-person shooter video game.

He also tweeted, “Shout out to the ROTC peeps.” A Facebook account shows he likes military-related sites.

Four women and one man, including a teenager and a senior citizen, were gunned down while shopping at a Macy’s department store in Burlington.

The Seattle Times identified one of the victims as Sarai Lara, a 16-year-old who survived cancer as a child.

A man who came to the door Sunday morning at an Oak Harbor address believed to be where Mr. Cetin’s stepfather and mother live asked an Associated Press reporter to leave the property.

Neighbor Jon Johnson, 26, said he believed Mr. Arcan still lived at the house.

“He lives directly behind us,” Mr. Johnson said. “We’ve seen him.”

Mr. Johnson said that when Mr. Cetin was arrested, “we were all like, ‘We’re pretty sure that’s him,’ and it ended up being him.”

Mr. Johnson said his household has had many disputes with Mr. Cetin’s stepfather.

“Two years ago on the Fourth of July, we were having a function with our friends, and [the stepfather] set tires on fire in the backyard,” he said.

Oak Harbor is a city of 22,000 on Whidbey Island. Many military families make their home in the city because Naval Air Station Whidbey Island is nearby. Oak Harbor is about 30 miles west of Burlington.

• This article is based in part on wire service reports.