Six bands were due to compete for a place on the pop group's 2021 tour, following weeks of pre-recorded audition episodes.
Perrie Edwards, Jade Thirlwall – both from South Shields – Jesy Nelson and Leigh-Anne Pinnock were not among those who tested positive.
The BBC One talent show sees the former X Factor winners create and mentor new bands.
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Its live show is now expected to air on Saturday October 24, the same date as Strictly Come Dancing's first live show of the series.
A statement from the BBC said: "We can confirm that a small number of people on the Little Mix The Search production have tested positive for coronavirus and they are now self-isolating following the latest Government guidelines.
"Due to the format of the show we have made the decision to postpone Saturday's programme.
"There are rigorous protocols in place to manage Covid-19 as the safety of all those involved in the production is paramount.”
The six bands have been living together with access to voice coaches, songwriters and producers who have also helped Little Mix in their career.
Little Mix formed on The X Factor in 2011 and have become one of the country's biggest-selling girl bands.
Recently, they announced details of their sixth studio album, Confetti, due for release on November 6.