Witnesses said a man carrying two knives was Tasered by police after running from the discount shop in Bury Street, Abingdon, at about 11:20 GMT.
Staff at a nearby Spar store said they barricaded themselves inside as the man tried to gain entry, before he was restrained by the police.
A 36-year-old man, who is from Abingdon, has been arrested on suspicion of murder.
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Police said the deceased was yet to be identified and his next of kin had not yet been informed.
Kash, who works at the Spar shop, said after the stabbing a man tried to push his way into the store.
He said: "He approached with two knives in his hands and he tried to enter. He said 'I just want to shake your hand' and we held the door closed.
"Then a policewoman caught up with him and Tasered him."
Poundland chief executive Jim McCarthy said he was "shocked" that one of the company's customers had been killed.
He said: "Our sympathy and thoughts centre on the family of the deceased and of course with other customers and colleagues who were in the store at the time of this terrible incident."
Supt Rory Freeman said: "Members of the public will understandably be shocked and upset by this incident today, which has happened in a very public place.
"At this stage, this is not being treated as a terrorist incident."
A second man nearby suffered a minor injury to his thumb. Thames Valley Police are investigating whether the incidents were connected.
Joanna posted on Twitter: "Watching everything that happened today in Abingdon from my office window was terrifying. Rest in Peace."
The leader of South Oxfordshire District Council John Cotton said on Twitter it was "dreadful news" and praised the quick response by police.
A local trader, who did not want to be named, said: "People are absolutely devastated. Normally it's a very busy vibrant market, but everyone has gone home."
Thames Valley Police has appealed for anyone who captured footage at the scene to come forward.
A customer has been stabbed to death in a Poundland store in Oxfordshire.