MU2B-40 pilot reported a brake malfunction on taxi.

Narrative

While taxiing after landing following a flight from ZZZ1 to ZZZ; the right brake was partially engaged and wouldn't release. I elected to continue to the FBO ramp rather than stop on the taxiway. Arriving at the FBO ramp; line personnel indicated that I had a small fire coming from the brake system. I deplaned and observed that the brake rotor was glowing red. The line personal put out the fire with a short burst from a dry chemical fire extinguisher.Once the rotor had cooled; I inspected the brake system and found that the paint had burned off the rotor and brake lining backing plates. I contacted both my maintenance and training providers who opined that the most likely cause was moisture in the hydraulic fluid which had frozen. They suggested that a small amount of fluid could be released from the bleeder valve which should free the brake. I pushed on the wheel and found that the brake had released by cooling without the need to open the bleeder valve.After refueling; I elected to depart ZZZ and fly to ZZZ2 my final destination. I did not use the brakes for landing or taxi but did apply them once on the ramp. They functioned normally.After putting the plane in the hangar; I noted that there was a small amount of hydraulic fluid seeping from the bleeder valve. I did not note any fluid at ZZZ.Upon reflection; I am not sure that; although the outcome was positive; the decision to depart ZZZ was the wisest. When I departed; ZZZ; I had a theory on the cause of the problem being moisture in the hydraulic fluid. Even though this was to prove to be correct; it was only a theory at the time of departure.Both flight conditions and human factors played a part in the mechanical issue and my decision making. The longest flight I have made in the airplane (about four hours) combined with the coldest temperatures (-42C) combined to produce the brake issue for the first time. Likewise; looking forward to returning home after a week away contributed to the decision to continue the trip. Not wanting to stop and block the taxiway contributed to my continuing to the ramp. Stopping and letting the brake cool would have prevented the brake from getting hot enough to burn paint.In the future; I would opt to stop on the taxiway rather than continue to the ramp. I would also obtain a sample of hydraulic fluid to confirm the hypothesis before departing.