The vessel is said to have been caught in a storm as it sailed towards Ushuaia in Argentina on Tuesday (29 November).
According to the BBC, quoting an Argentinian media report, the victim was a US woman who died after being struck by shattered glass.
Viking said it was investigating the incident and offered its "deepest sympathies" to the passenger’s family.
"Our focus remains on the safety and wellbeing of our guests and crew, and we are working directly with them to arrange return travel," the company added.
The ship, which was completed earlier this year, arrived in Argentina on Wednesday (30 November) and had sustained "limited damage".
As a result of the incident, the company has cancelled its scheduled voyage from Antarctica between 5-17 December.
