SAN ANTONIO — Donald J. Trump got Brenda Hernandez’s hopes up in 2016 when he indicated that he had a heart for young undocumented immigrants like her, who were brought into the country by their parents. But her hopes were dashed when Mr. Trump became president and ordered an end to the federal program known as DACA that shields her from deportation.

After the president’s latest announcement on Saturday — that he was proposing to end the standoff over the partial government shutdown by, among other things, temporarily extending that shield — Ms. Hernandez just shrugged.

“I don’t trust Trump — I don’t believe him,” she said as she hoisted her 3-year-old son onto a swivel chair at the D & H Beauty Salon in San Antonio. The blue-collar shop, with rows of wooden chairs in the waiting area, is a favorite among immigrants in the city. But in the hours after Mr. Trump’s White House address, his latest offer of an immigration deal was finding little support.

[Read: In Trump’s Immigration Announcement, A Compromise Snubbed All Around]

In exchange for $5.7 billion to erect a barrier along the border with Mexico, Mr. Trump said he would agree to extend protection for three years for the roughly 800,000 immigrants who benefit from the Deferred Action for Childhood Arrivals program, which covers people brought into the country illegally as children. The president held out the hope of similar relief for 300,000 immigrants living in the United States with temporary protected status, known as T.P.S., after fleeing earthquakes and other disasters in their own countries.