It is quite possible for an aeroplane to have its best angle of climb with flap deployed, although almost certainly at a lower speed than clean.
The increased parasite drag of deploying the flaps is relatively low due to the low speed, and by changing the aerofoil section induced drag may well be reduced. This results in a new drag curve with a new lower VMD. The drag at this speed may or may not be less than the drag at the clean VMD, but coupled with the increased thrust mentioned by HAWK37 it is quite possible for the speed of maximum excess thrust (VX) to have a greater excess available than at the clean VX.
OLD SMOKEY asks why if this is the case we do not leave flaps out permanently? Quite simply because the parasite drag increase with speed is very rapid compared to a clean aircraft, therefore there is a serious penalty on cruising speed. In addition there are all sorts of structural complications from the lift distribution at high speed, hence we have VFE limits dramatically lower that VNE/VMO.
As for examples, a quick perusal of the performance section of the flight manual of any Citation (and many other light jets) gives the result that the second segment climb gradient (Flaps 15, gear up) is often significantly better than the (clean) final segment gradient for the same conditions.
I should clarify this by saying that the "often" is at low altitudes and temperatures; as it gets "hot & high" the reducing thrust available and TAS effects make it revert to the intuitive situation.
Have a good Xmas!