The government snared more than 3,000 immigrants who illegally attempting to cross into the U.S. in just one day last week, the Trump administration’s top border official told Congress on Tuesday, saying the situation qualifies as a full-blown “crisis.”

Kevin K. McAleenan, commissioner of U.S. Customs and Border Protection, said the majority were traveling as families and another 350 were children traveling without parents.

Between those and the children who came as part of families, that means more than 1,100 juveniles caught that day were forced to make a treacherous journey he said that often involves sexual abuse and other trauma.

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“It is indeed both a border security and a humanitarian crisis,” he told the Senate Judiciary Committee.

The 3,029 people caught Dec. 3 is the highest one-day total in years, and he said at the current rate, the country would shatter last year’s record number of immigrant families who attempted to push their way into the country without permission.

More than 2,500 of the people caught that day were snared by Border Patrol agents while they jumped the border. The rest showed up at border crossings and demanded entry despite lacking permission.

One of those caught jumping the border was Nicolas Pablo, who was part of a group of 13 people agents found after they jumped the border fence and tried to hide in the sandy washes and bushes in the desert near Calexico, California.

Agents snared Mr. Pablo about 70 yards north of the border, and he said he was from Guatemala and had no right to be in the U.S.

He was charged with illegal entry, a misdemeanor.

An hour’s drive to the east, near the divide between California and Arizona, agents spotted a group of four immigrants climbing over the wall near the Andrade border crossing.

Members of that group also are being prosecuted for illegal entry.

But the vast majority of the 3,000-plus immigrants encountered that day are not facing prosecution.

Indeed, the children and families — almost all of them likely to be from Central America, the source of recent years’ surge of illegal immigration — have likely been released into the country to await deportation hearings or asylum cases.

Few will qualify for relief — but if data from the last two years is a guide, few of them actually will be deported anytime soon. Many will ignore the courts and disappear into the shadows with the 11 million other immigrants already here without permission.

Mr. McAleenan said that while arrests on Dec. 3 were particularly high, his agency is averaging at least 2,000 entries a day.

“This snapshot is emblematic of the steady trends we’ve been seeing for many months,” he said.