Based on current evidence, Georgia has the oldest flag still in use as a nation-state.
It is believed that Georgia first adopted their current flag design, featuring a white field with a central red St. George's cross and four smaller red crosses in the quadrants, in 1008 A.D. This predates the Danish flag, the Dannebrog, which is often cited as one of the oldest, though its definitive use is documented later. While the Georgian flag experienced periods of disuse during Ottoman and Russian rule, it was officially reinstated in 2004.
The history of flags can be difficult to precisely trace, making definitive claims of "oldest" complex. However, given the documented use dating back to 1008 A.D., Georgia's flag appears to be the oldest national flag still in use today.