FlyBe allow you to take a single bag on board, free of charge, as long as it doesn't weigh more than 10kg or have dimensions exceeding 55 x 35 x 20cm.
You can also have a smaller additional bag (for under-seat storage), such as a handbag, as long as the combined weight of the two bags doesn't exceed 10kg.
http://www.flybe.com/baggage/hand-baggage.htm
That's basically the same as many other airlines, Ryanair, for example, has basically the same rule except that the larger bag can be 55 x 40 x 20cm (and they impose specific dimension limits on the smaller bag):
https://www.ryanair.com/gb/en/useful-info/help-centre/faq-overview#1-1
BA's size restriction for the larger bag is slightly more generous (at 56 x 45 x 25cm) but not that much. Jet2 also has the same size limit.
FlyBe is the only operator to Bergerac from Southampton. The only other airlines flying from the UK to Bergerac are BA (from London City, starting next May), Jet2 (from Leeds/Bradford, seasonally) and Ryanair (from Stansted, with additional seasonal services from Bristol, East Midlands and Liverpool).