5 July 1934: birth of Philip Madoc, who is Brockley in Daleks - Invasion Earth 2150 AD (Movie 1966), Eelek in The Krotons (Second Doctor, 1968-9), the War Lord in The War Games (Second Doctor, 1969), Solon in The Brain of Morbius (Fourth Doctor, 1976), and Fenner in The Power of Kroll (Fourth Doctor, 1978-9).
5 July 1970: birth of Toby Whitehouse, who wrote School Reunion (Tenth Doctor, 2006); Greeks Bearing Gifts (Torchwood, 2006); The Vampires of Venice (Eleventh Doctor, 2010); The God Complex (Eleventh Doctor, 2011); A Town Called Mercy (Eleventh Doctor, 2012); Under the Lake / Before the Flood (Twelfth Doctor, 2015); and The Lie of the Land (Twelfth Doctor, 2017)
ii) broadcast anniversary
5 July 2008: broadcast of Journey's End, the last episode of Season 4 of New Who, with two Tenth Doctors, Donna Noble, Jack Harkness, Rose and Jackie Tyler, Sarah Jane Smith, Martha Jones, Mickey, Gwen, Ianto, Luke, K9 and the Daleks and Davros. After the brilliant cliff-hanger of the previous week, when it looked as if Ten was about to regenerate, I was really disappointed by this, though we had had fair warning that RTD couldn't really manage his own finales the previous year. The bit that worked best for me was the dramatic and tragic end to Donna's story, though I know others are more attracted to Handy and Rose heading off together into the sunset.
iii) date specified in-universe
There is a street in Buenos Aires called "5 de Julio", visible on a map shown to CIA officers in the last 2011 Torchwood episode, The Blood Line. (It's the day that an invading British force was defeated in 1807. The street map shown here bears only a loose resemblance to the real map of Buenos Aires, though there is a real street called 5 de Julio.)