Grigg's departure comes months after the company, which owns London's Broadgate and Paddington Central complexes, posted a staggering 1.1 billion pound ($1.43 billion) annual losses in May as the outbreak shut down thousands of shops and forced some chains into administration.
Carter, who has been the chief financial officer at the company since 2018, will take over from Grigg on Nov. 18. Carter's successor will be announced in due course.
Shares in the company, which owns a range of office-led London campuses and retail spaces, were flat in early trading after having plunged 44.60% so far this year.
The pandemic has already forced British Land rival Intu into administration earlier in the year, while Hammerson last month announced plans to raise 825 million pounds to weather the crisis.
British Land has also been struggling with a slump in the value of its properties and in May conducted a series of rent deferrals and cancellations to help its tenants ride out the crisis.
(Reporting by Samantha Machado in Bengaluru; Editing by Rashmi Aich)