I had two Mini Countrymen and a Mini Clubman Estate and all three had the battery under the back seat. There was a little vinyl cover to go over the terminals to stop them from touching the steel frame under the rear seat. If you forgot to put the cover back on, or if the cover had been damaged (by being squashed between the terminals and the seat frame) then the smoke and smell quickly alerted you to the problem.
The first time it happened to me, the steel frame work had heated up enough to char and smoulder the plywood seat base which sat on the frame.
Worse still, to remove or replace the battery you had to tip the front seat forwards, lift up the front-hinged rear seat and rest it against your legs, then bend right over it to lift or lower the battery, back aches GUARANTEED!