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.
Updates on this story and more from Oxfordshire
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.

Anthony Maslin and Marite Norris lost their children Mo, Evie and Otis, along with Ms Norris's father Nick Morris, when the flight came down in eastern Ukraine in July 2014.
The Perth couple described the birth of Violet May Maslin as an "amazing gift".
"Violet's birth is a testament to our belief that love is stronger than hate," they said in a statement.
"We still live with pain, but Violet, and the knowledge that all four kids are with us always, brings light to our darkness.
"As Martin Luther King said, 'Darkness cannot drive out darkness; only light can do that. Hate cannot drive out hate; only love can do that'."
Malaysia Airlines Flight MH17 was shot down as it flew over conflict-hit Ukraine from Amsterdam to Kuala Lumpur.
Mr Maslin and Ms Norris had stayed in Amsterdam while their children, aged eight, 10 and 12, flew back to Australia with their grandfather.
A total of 298 passengers and crew were killed, including 80 children in total.
The flight was hit by a Russian-made Buk surface-to-air missile.
Western officials and Ukraine's government blame Russian-backed rebels for shooting down the plane.
But Russian authorities have consistently denied responsibility, instead blaming Ukrainian government forces.
A customer has been stabbed to death in a Poundland store in Oxfordshire.
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An Australian couple whose three children died when Flight MH17 was shot down have a new daughter.