The case is not linked to the scandal at the German migration agency, but together they have played into fears that Ms. Merkel’s government is not in control of the migrant situation.

The scandal at the migration agency has been building for weeks since state prosecutors began a criminal investigation into alleged corruption at a regional office in Bremen, in northern Germany. The investigation was recently broadened to include at least 10 other offices with above-average rates of granting asylum.

Critics of Ms. Merkel, led by the nativist Alternative for Germany party, now the largest opposition party, have been calling for a parliamentary inquiry into her migration policy. The proposal is gaining traction with other parties, too, and could threaten Ms. Merkel’s uneasy coalition with the Social Democrats.

On Thursday night, after reports of the killing, Alice Weidel, the deputy leader of the AfD, accused Ms. Merkel of sharing responsibility in the death of the girl, who has been identified by the authorities as Susanna Feldmann, and called for her entire cabinet to step down.

“Make way for an asylum policy that is built around law and order, so fathers and mothers in our country no longer need to be afraid for their children,” Ms. Weidel said in a video posted on Twitter.

“What do you say to the parents of the murdered #Susanna, Frau Merkel?” she tweeted later.

But there was plenty of outrage among centrist politicians, too. “Why was the perpetrator able to leave the country apparently under a false name?” asked Christian Lindner, the leader of the liberal Free Democrats, who has also expressed support for a parliamentary inquiry.