"We mourn the loss of a dear colleague, a leader and highly valued member of the senior leadership team," CEO Neal Froneman said.
"He will be missed."
The Minerals Council South Africa, where Bessit was a board member and CEO of the Zero Harm Forum, also offered its condolences.
"He was a dedicated and compassionate leader and played a material role in the minerals council and in the mining industry," the council said.
"Having spent three decades working in the mining industry, Shadwick was deeply committed to the South African mining industry and showed great concern and empathy for those working within the industry and living in the surrounding communities."
Bessit had joined Sibanye-Stillwater on its formation in 2013 as Sibanye Gold, following the unbundling of Gold Fields' Kloof, Driefontein and Beatrix gold mines.
He had previously worked at Gold Fields, at Impala Platinum as general manager and at AngloGold Ashanti, where he moved through the ranks to general manager level at the Deelkraal, Elandsrand and Savuka mines.
He leaves behind his wife, three children and two grandchildren.
Sibanye-Stillwater said William Osae, SVP technical services for its South Africa gold operations, had been appointed head in the interim in order to ensure operational continuity and the ongoing focus on safe production.
The number of COVID-19 deaths in the country's mining industry has risen to 244 and the number of active cases stands at 1,822, according to yesterday's figures from the minerals council.
Cases in the country have surged as it battles a variant strain of the virus.
South Africa has a cumulative total of 1.35 million cases and more than 38,000 deaths, according to Johns Hopkins University today.