A Panama hat was a type of hat.
The First Doctor, (TV: The Chase, The Daleks' Master Plan) Fifth Doctor, (TV: Castrovalva) Seventh Doctor (TV: Time and the Rani) and Eleventh Doctor all possessed a kind of Panama hat, (COMIC: The Dalek Project, PROSE: Chasing the Dawn) as did a parallel universe Doctor. (PROSE: Sustainable Energy) The Eleventh Doctor noted that "Panama hats are cool." (COMIC: The Dalek Project)
The Eleventh Doctor once wore a Panama hat that was slightly too small on him, whilst he relaxed with a book on a hammock hung from the TARDIS on the International Date Line. (PROSE: Chasing the Dawn)