Angela Merkel, Germany's chancellor, attends a news conference at the Chancellery in Berlin on Feb. 1. (Krisztian Bocsi/Bloomberg)

She was Time magazine’s Person of the Year, a compassionate leader who opened Germany’s door to more than a million desperate migrants. Frau Nein became Frau Nice. There was even talk of a Nobel Peace Prize.

But German Chancellor Angela Merkel is suffering a harsh reversal of fortune, confronting a political backlash that is isolating her both at home and across Europe. As Merkel is pushed into a corner on migrant policy, political pundits are sounding a once-unthinkable alarm, warning that her job may be at risk if she does not quickly change course.

“I don’t think there is any question anymore,” said Werner J. Patzelt, a political analyst at Technical University Dresden. “Angela Merkel is really in trouble.”

For Merkel, the bad news just keeps getting worse. In the aftermath of attacks in Cologne on New Year’s Eve — in which asylum-seekers allegedly assaulted dozens of German women — a new poll found that 40 percent of respondents now want her to resign. Rebel lawmakers in her ruling coalition are openly criticizing her. The head of the Christian Social Union in Bavaria — formerly a staunch ally — is even threatening to sue the government if it does not curb the influx.

A new German poll says 81 percent of those asked think the government mishandled the refugee crisis and Merkel’s approval rating has fallen to 46 percent, the lowest since August, 2011.

Known for ruling by opinion poll, Merkel has seemed to backtrack on aspects of her open-door policy in recent days — insisting, for instance, that most people seeking refuge in Germany should go back home after peace comes to countries such as Syria and Iraq.

Her cabinet on Wednesday backed new measures aimed at delaying refugees from bringing in close relatives for 2 years and declaring three North African countries as ‘safe,’ making it far harder for asylum seekers from those countries to win refugee status

But she is still mostly sticking to her guns and refusing to close Germany’s doors. It is presenting a chancellor who first came to power when George W. Bush was still the U.S. president with one of the toughest choices of her decade-long tenure: whether to keep holding up the banner of humanitarianism or to be politically expedient.

“Merkel has become a prisoner of her own politics,” said Jürgen Falter, a political scientist at Mainz University. He added, “I think the likelihood is about 60 percent that her policies don’t work out and she throws in the towel.”

It is an unusually tight spot for the Iron Chancellor, a woman who rose to be the de facto leader of Europe by driving hard bargains on rescues for bankrupt Greece. In the process, she elevated Germany to the zenith of its post-World War II power.

But the refugee crisis has damaged her profoundly, underscoring the high price of compassion in a risk-averse world. A nation whose World War II past made it fully aware of the dangers of xenophobia, modern Germany was leading by example in the 21st century, becoming a beacon of hope for desperate foreigners fleeing war and poverty. Merkel staked her job on upholding what she likes to call “European values” — in effect, that the progressive people of wealthy Europe should not turn their backs on the human right to sanctuary for Syrians, Iraqis and others.

But she has run into serious stumbling blocks. The attacks in Cologne did not help her cause, nor did the November massacre in Paris that occurred after militants entered Europe disguised as migrants. Additionally, a large percentage of the new comers, it turns out, were not really escaping war at all — but seeking to leverage German kindness to build lives away from places such as North Africa, the Balkans and Pakistan.

She also erred by effectively promising her countrymen something that she has thus far been unable to deliver: A pledge that other nations in Europe would take in more migrants and start to share Germany’s burden.

Instead of pitching in, countries across Europe are barring their doors. A voluntary European program to legally resettle refugees has failed, with nations mostly refusing to accept newcomers from the Middle East and elsewhere.

In fact, both publicly and privately, European politicians long opposed to welcoming refugees are reveling in the schadenfreude of Merkel’s comeuppance. Last week, Czech Prime Minister Bohuslav Sobotka announced that a bloc of anti-refugee nations — also including Hungary, Poland and Slovakia — would hold their own meeting ahead of a key E.U. summit in February to discuss alternative solutions to the crisis. Suggesting Berlin has gone too soft, Slovak Prime Minister Robert Fico scoffed that migrants have become a “protected species” in Germany.

Meanwhile, the EU finally approved a key $3 billion deal on Wednesday in which Turkey would crack down on the human traffickers ferrying thousands of migrants to Europe via Greece every week. But Turkish demands for more money raises questions about how quickly change may happen on the ground.

Add it all together, and Merkel is in a precarious spot. If she sticks to her principles, it means Germany stands virtually alone in Europe as a haven for migrants. That is a burden that the Germans — initially welcoming to the waves of refugees — are increasingly reluctant to shoulder.

Yet Merkel also can take stock in a few things going in her favor. Wolfgang Schäuble, her famously strict finance minister, has been floated by some as a possible successor. But insiders say he may be reluctant to steal the job from Merkel and is not obviously angling for the post. And given her still formidable influence, any move by her critics for a no-confidence vote in Parliament remains an uphill battle for now.

However, if her Christian Democrats score a resounding defeat in the March local elections, the pressure on Merkel to change course or step aside could grow to a clamor. But the question then would be whether Merkel is willing to go as far as closing German borders — something that could put a nail in the coffin of the open-border policies once hailed as the single largest accomplishment of the European Union.

“She is extremely worried about the state of the public mood, but she also sees a bigger picture,” said Stefan Kornelius, foreign editor of the Süddeutsche Zeitung. “It feels like she is fighting for the European soul.”

Stephanie Kirchner in Berlin and Karla Adam in London contributed to this report.

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