A 21-year-old white man was arrested at the Walmart store where the attack took place in the city of El Paso, near the US-Mexico border, on Saturday.
A document, apparently posted shortly before the attack and believed to have been written by the man, espoused white nationalist and racist views.
The tract called the attack a response to "the Hispanic invasion of Texas".
The man lived in Allen, in the Dallas area, about 650 miles (1,046km) east of El Paso, according to police, who were still interviewing him. It is not yet clear how long before the attack he had gone to the city.
Officials have not offered a precise motive for the attack, which also left 26 injured, but El Paso Police Chief Greg Allen said investigators were looking into a document that could indicate a "potential nexus to a hate crime".
He appeared to be referring to a text posted on 8chan, an online message board frequently used by the far right, which describes a "cultural and ethnic replacement brought on by an invasion", alluding to Hispanic people in the US.
The four-page document, reportedly posted some 20 minutes before police received the first emergency call, also expresses support for the gunman who killed 51 people in Christchurch, New Zealand, in March.
The man - named by US media as Patrick Crusius - opened fire with an assault-style rifle and surrendered after being confronted by officers outside the store, police said.
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"We're going to aggressively prosecute it both as capital murder but also as a hate crime, which is exactly what it appears to be," Texas Governor Greg Abbott told reporters. He called it "one of the most deadly days in the history" of the state.
The attack
The shooting, believed to be the eighth deadliest in modern US history, took place in a city where most of the population of 680,000 is of Hispanic descent. It lies along the Rio Grande across the border from Ciudad Juárez in Mexico.
The victims have not yet been named but Mexico's President Andrés Manuel López Obrador said three Mexican nationals were among the dead.
CCTV images said to be of the attacker show an armed man in a dark T-shirt wearing eye glasses and what appears to be ear protectors.
Image copyrightAFP
Image caption
CCTV images said to be of the gunman have emerged
Reports of an active shooter were received at 10:39 local time (16:39 GMT), and law enforcement officers were on the scene within six minutes, Mr Allen said.
The Walmart, near the Cielo Vista Mall, was full of shoppers buying back-to-school supplies at the time of the shooting and witnesses described scenes of chaos as customers fled for their lives.
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"People were panicking and running, saying that there was a shooter," Kianna Long told Reuters news agency. "They were running close to the floor, people were dropping on the floor."
Ms Long said she and her husband ran through a stock room before taking cover with other customers.
The attack came a few hours before another mass shooting in Dayton, Ohio and six days after a teenage gunman killed three people at a California food festival.
The reaction
US President Donald Trump called the attack "an act of cowardice" on Twitter: "I know that I stand with everyone in this country to condemn today's hateful act. There are no reasons or excuses that will ever justify killing innocent people".
Democratic presidential candidate Beto O'Rourke, an El Paso native, abandoned a campaign event in Las Vegas to return to his home town. Earlier, he told a labour forum the attack had shattered any illusion that gun reform will "come of its own accord" in the US.
New Jersey Senator Cory Booker, another Democratic presidential hopeful, said the US seemed to be "accepting the idea that these [shootings] are going to be a regular occurrence".
El Paso's Democratic congresswoman, Representative Veronica Escobar, told the New York Times the attack was "about hate".
Media captionAfter the Las Vegas attack in October 2017 the BBC looked at how US mass shootings are getting worse
The shooting refreshed calls for gun control in the US but Texas Attorney General Ken Paxton said it would probably not have stopped the attack.
He added that if a "crazy" gunman launched such an attack, there was no way that law enforcement officers could be there to stop it, telling CBS News: "The best way is to be prepared to defend yourself."
The attack is the second deadly shooting to take place in a Walmart store this week, after an ex-employee of the company killed two former co-workers at a Mississippi branch on Tuesday.
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