According to Collins English Dictionary one usage of the word puss is an informal name for a hare (related to Middle Low German, Dutch and Lithuanian, apparently). I do remember hares being referred to as puss, or pussy, by some rural Scots around fifty years ago.
Rabbits were coneys, from the Gaelic name for them 'coineen' (with my apologies to the Gaeltacht for the phonetic spelling).



BTW, Mrrzy, the population of the UK is British while the language remains English. To add to the confusion the British Isles includes the completely separate country of Southern Ireland, whose people speak English and Irish.