Disclaimer: the date and nature of this session is far from certain. What follows is a current 'best guess' based on available evidence, but any corrections or further comments are welcome.
Photographs by Ethan Russell published in the Get Back book (which accompanied the Let It Be box set in 1970) and the hardback book from the 2012 vinyl box showed The Beatles recording in a session in the studio basement at Apple's headquarters. However, comparisons with the film footage and other photographs by that time show that the images cannot have come from the Get Back/Let It Be sessions in January 1969 - the main giveaway is the length of head and facial hair of the musicians.
Following the Get Back/Let It Be sessions in January 1969, The Beatles reconvened at Trident Studios on 22 February to work on I Want You (She's So Heavy). This session is believed to have taken place before that date.
The Beatles are known to have re-recorded parts of Don't Let Me Down at an undocumented session; the vocals on the session tapes and the single are different, with an extra overdub having been added sometime after January 1969. John Lennon and Paul McCartney both corrected elements of their vocals parts, with Lennon double-tracking his during the middle eight and ending.
Although far from certain, it appears likely that this session was a rehearsal for I Want You (She's So Heavy), in which the new Don't Let Me Down vocals were also recorded. Additionally, a typewritten sheet next to Ringo Starr in one photograph looks like the lyrics for Get Back, although that shouldn't imply that The Beatles performed it on this day.
Dating it precisely has proved problematic, but it is known that George Harrison was in hospital from 7-15 February having his tonsils removed, and hair length indicates this session took place after that time. Starr was filming The Magic Christian during this period, although he was free on most evenings and weekends.
Glyn Johns and Billy Preston had been in America during mid-February, returning in time for the Trident session on Saturday 22 February. It is therefore possible that The Beatles reconvened for a rehearsal a day or two beforehand, to best prepare the song and minimise timewasting in the studio - a practical consideration following the drawn-out sessions for the Get Back project.