0915 GMT December 14, 2018
Beatrix von Storch, deputy leader of the party's parliamentary faction, had criticized Cologne police for sending a New Year's greeting in Arabic on Twitter, AFP wrote.
"What the hell is going on with this country? Why is an official police site... tweeting in Arabic?" she wrote. "Did you mean to placate the barbaric, Muslim, gang-raping hordes of men?"
Von Storch's tweet appeared to reference chaotic 2015 New Year's Eve celebrations in Cologne, which were marred by mass sex assaults on women by men of mostly North African origin.
Cologne police said their complaint alleges that the tweet flouted laws against incitement to hate.
Separately, prosecutors in the western city said they had received hundreds of complaints over the AfD lawmaker's comment.
Von Storch's account was blocked by Twitter for 12 hours following her post, and the tweet has since been removed.
She posted the same comment on Facebook, which then also took it down, citing "incitement to hate (paragraph 130 of the German penal code)".
Twitter and Facebook's tough stance came as an anti-online hate speech law came into effect on January 1 in Germany.
Social media companies that fail to remove illegal inflammatory comments could face up to 50 million euros ($60 million) in fines.
AfD chief Alexander Gauland took aim at the new rules, saying in remarks carried by national news agency DPA that the regulations are "Stasi methods that remind me of communist East Germany".
Gauland, who is not active on social media, encouraged users of such websites to "keep publishing" the erased comments from von Storch.
A Cologne police spokesman said it was long-standing practice to send out information in several languages during large gatherings and events.
"We simply want people to be able to understand us," said the spokesman.
The AfD seized 92 parliamentary seats in September elections — the strongest showing for a far-right party in the post-war era — as it capitalized on discontent over the more than one million asylum seekers who arrived in Germany since 2015.