Police have not yet identified him or the stabbing victims, although they said the person who died was 50 years old.
- Who was the attacker?
Officers in the northern port city's Barmbek district said the attacker has been arrested.
Those hurt in the supermarket included a 50-year-old woman and four men aged 19 to 64, while a 35-year-old Turkish man was wounded while capturing the attacker. Shortly afterwards, police officers arrived at the scene and arrested the suspect.
The motivation for the attack remains unclear, but a video posted on social media showed an eyewitness claiming that the attacker stabbed three people before shouting "Allahu Akbar", or God is great in Arabic.
Mr Froehlich said authorities intend to ask that he be held in custody on suspicion of murder and five counts of attempted murder, but may seek to have him held at a psychiatric unit instead. Police said on Friday the man was born in United Arab Emirates. "First reports about a possible robbery can not be confirmed so far", he said.
"I am furious that the offender is apparently someone who has claimed protection with us in Germany and then directed his hate against us", the mayor added. "They said he started talking a lot about the Quran, stopped drinking alcohol and questioning many things". Der Tagesspiegel, a daily based in Berlin, said the suspect was of Palestinian ethnicity and had come to Germany seeking asylum.
On July 24, 2016, a 27-year-old failed Syrian asylum seeker blows himself up near an open-air music festival in the southern city of Ansbach, wounding 15 others.
The similarity between the two cases risks reopening barely healed wounds over Chancellor Angela Merkel's decision to allow more than million migrants into Germany since 2015, with just two months to go until legislative elections in September.
But "people were running after him with improvised weapons, chairs and sticks. they ran after him into a side street", a witness Ralf W told Bild newspaper.