In March, a federal investigator asked the New York Police Department for information about a woman who had been arrested during a pro-Palestinian protest at Columbia University and was now detained for overstaying her visa.

The woman, Leqaa Kordia, 32, was being investigated for money laundering, the investigator said, and the Department of Homeland Security needed help. The police handed over her birth date, address and the name of a possible associate. An officer also provided the woman’s sealed arrest report.

But a month later, during an immigration court hearing, the only evidence of money laundering that federal prosecutors presented was a $1,000 MoneyGram transfer that Ms. Kordia had sent to relatives in Gaza.

The judge, Tara Naselow-Nahas, was unimpressed.

“Based on the evidence, I do not find that the respondent poses a danger to the United States,” she said and ordered that Ms. Kordia be released on a $20,000 bond. Ms. Kordia remains at the Prairieland Detention Facility in Texas as prosecutors seek a reversal of the decision.

But the judge’s ruling and questions about the federal government’s credibility have civil libertarians asking whether the Police Department should reconsider its cooperation with the Trump administration.

The city’s sanctuary laws forbid the department from divulging information in immigration cases, which are civil matters, but the police often cooperate with federal authorities on criminal cases, usually in joint investigations into crimes like sex trafficking, drug and gun dealing, and terrorism.