I have an adored drill/driver that has always struggled with some cheap bits I had from another kit. Drilling into wood goes as planned, but then when pulling the bit back out of the wood, the bit always slips out of the chuck, even when I turn the mode to reverse. I always assumed it was the cheap bits, but after I purchased a very nice set of hex bits, I am experiencing the same issue. The confusing part of the situation is that fastener bits never come out. I just compared the base of the drill vs fastener bits and the inner neck of them seem to be slightly larger on the drill bits. Could the ball bearings in the chuck not be getting a good enough bite? This is such a small issue, but it is extremely annoying. I don't think I have used the drill enough for wear.
Drills do not work in reverse
The reverse setting on the tool is for use with screws and bolts, and maybe for taps if you have a power-friendly tapping situation. It is not for use with drills.
Drills are only made to go one way, ever.
When withdrawing a drill from a hole, you are to continue running the drill in the forward direction, and pull it out of the hole while it is spinning forward. It is normal for the drill to start to bind as you pull it out. The sharpened flutes of the drill will make short work of whatever is binding it. The spiral will continue to pump material out of the hole.
If you run the drill backwards, there is no cutting edge there, and so cruft and debris tends to be plowed into the space between drill and the wall of the hole. This quickly causes the drill to bind.
With the drill binding in the hole, pulling the tool out will cause the drill to come out of the "hex" quick release in your tool. It is not designed to firmly clamp the bit, it is designed for quick release of screw bits.
The drill would behave similarly in a proper chuck, by the way, but a chuck has much more gripping power.