The WikiLeaks founder is battling against attempts to send him to the US for trial for allegedly conspiring to hack government computers and violating an espionage law over the release of confidential cables in 2010 and 2011.
Judge Vanessa Baraitser adjourned the case until Monday after being told one of the lawyers involved had been exposed to the virus.
She told London’s Old Bailey that the lawyer was being tested on Thursday, with the result due on Friday.
“At the moment we would respectfully submit we have to go ahead on the assumption that she has Covid,” Edward Fitzgerald, Mr Assange’s lawyer said.
“If that is the correct assumption … we shouldn’t really be here. Covid would be here in the courtroom and it’s not possible to tell how far it’s extended.”
It comes amid safety concerns over the resumption of criminal trials and in-person hearings in many courts across England and Wales.
Birmingham Crown Court was evacuated on Tuesday and closed temporarily for a deep clean, aftera member of staff developed coronavirus symptoms.
In August, Manchester Crown Court was shut after a Covid-19 outbreak among staff.
The Ministry of Justice has been attempting to increase the number of hearings after the backlog of criminal cases passed 500,000 in England and Wales.
Delays have resulted in a controversial law change to allow suspects to be held in custody longer before trial.
Mr Assange has been held on remand at HMP Belmarsh for a year after serving a 50-week jail sentence for breaching bail conditions by fleeing to London’s Ecuadorian embassy in 2012.
His current extradition hearing, which was previously delayed by the coronavirus outbreak, is expected to last for around a month.
Mr Assange is fighting extradition to the US on an 18-count indictment, which alleges he plotted to hack computers and conspired to obtain and disclose national defence information.
If convicted, he faces a maximum possible penalty of 175 years in jail.
Mr Assange’s lawyers have argued that he should be granted bail because of the pandemic, as he has suffered from respiratory infections and heart problems.
A judge has ordered him to be kept in prison because he is considered a flight risk.
Wednesday’s hearing was told that the Australian’s political opinions put him in the “crosshairs” of the Donald Trump administration.
Professor Paul Rogers told the Old Bailey that WikiLeaks’ revelations were “significant” in showing how the US coalition’s wars in Afghanistan and Iraq were “going wrong” despite public claims of their success.
Julian Assange in the Ecuadorian embassy – a timeline
The emeritus professor of peace studies at Bradford University said leaked documents published by WikiLeaks exposed details of an extra 15,000 civilians killed in Iraq.
Asked by Mr Fitzgerald QC if he believed the charges against Mr Assange are motivated by “criminal justice concerns” or political considerations, Prof Rogers said: “I have to say it appears to be the latter. This does appear to be a political trial”.
The defence witness said the fact president Barack Obama’s administration did not prosecute Assange was also a factor in the Trump administration’s decision making.
Trevor Timm, co-founder and executive director of the San Francisco-based Freedom of the Press Foundation, said his organisation had contributed around $100,000 to Mr Assange’s legal costs.
He described the case as “a dire threat to press freedoms in the US” and agreed that it was the “thin end of the wedge to prosecute journalists”.
Additional reporting by agencies