The Namibian presidency in a social media post said Geingob's medical team at Lady Pohamba Hospital did its best to help him but could not save the president.
He is survived by his wife, Monica Geingos, and three children, who were by his side during the last hours.
Geingob was undergoing cancer treatment and had a colonoscopy and a gastroscopy on 8 January following a biopsy. His office said he was supposed to travel to the US for treatment and return to Namibia on 2 February.
Acting president Angolo Mbumba on Sunday called for calm, saying that the "cabinet will convene with immediate effect in order to make the necessary state arrangements in this regard".
“The Namibian nation has lost a distinguished servant of the people, a liberation struggle icon, the chief architect of our constitution and the pillar of the Namibian house,” Mr Mbumba said in a statement on social media.
Geingob took office as the third president of Namibia in 2015 and had earlier been the longest-serving prime minister of the South African nation. He was due to finish his second and final term in office later this year.
Born in 1941, Geingob became the first prime minister of Namibia after it gained independence from South Africa in 1990.
Throughout his political career, Geingob suffered multiple health-related issues.
In 2014, he said he had survived prostate cancer just a year after undergoing brain surgery. In 2023, he reportedly underwent an aortic operation in neighbouring South Africa.
Geingob led Namibia to support South Africa's complaint against Israel's barbarity in Gaza at the International Court of Justice under the Genocide Convention. Namibia also issued a scathing criticism of its former colonial ruler Germany for defending Israel.
South African president Cyril Ramaphosa in a statement said: "Today, South Africa joins the people of our sister state Namibia in mourning the passing of a leader, patriot and friend of South Africa.
Today, South Africa joins the people of our sister state Namibia in mourning the passing of a leader, patriot and friend of South Africa.
Our thoughts and prayers are with the Geingob family and the people of Namibia who have lost an outstanding leader in a year in which… pic.twitter.com/ssn7axOa3s
— Cyril Ramaphosa 🇿🇦 (@CyrilRamaphosa) February 4, 2024
"President Geingob was a towering veteran of Namibia's liberation from colonialism and apartheid. He was also greatly influential in the solidarity that the people of Namibia extended to the people of South Africa so that we could be free today."
Kenya’s president William Ruto said Geingob was a “believer of a unified Africa and strongly promoted the continent's voice and visibility at the global arena”.
“May God give the people of Namibia strength and courage during this difficult period,” he added.
I extend my deepest condolences and sympathy to the family and the people of Namibia following the death of President Hage Geingob.
President Geingob was a distinguished leader who served the people of Namibia with focus and dedication.
He was a believer of a unified Africa…
— William Samoei Ruto, PhD (@WilliamsRuto) February 4, 2024
Namibia will hold presidential and assembly elections in November this year.