In 1992, Gunn's two-year-old daughter, Francesca, was diagnosed with leukaemia. Norwich City fans were at first astonished by the sight of the goalkeeper running out with a completely shaven head. In the words of author and Norwich supporter, Kevin Baldwin, "Occasionally, the gap between the crowd and the players can cause unfortunate misunderstandings. A few months ago, Gunn shaved his head and we all laughed at him ... I was especially keen ... to shout "Baldy!" ... It now turns out that his daughter was undergoing chemotherapy at the time, which made her hair fall out. He shaved his head to show her that this was nothing to be ashamed of. Sorry, Bryan." When news spread of the reason for Gunn's shaven head, he received "support from the whole of the sporting world and the people of Norfolk".