In reply to SFM:
Muskrat. Escaped and bred in both England, Scotland and Eire, in the 1920s, but after a campaign of trapping & shooting all the existing Muskrats had been eradicated by the late 1930s.
Black tailed Prairie dogs have bred in the wild in the UK but it is believed those too have been eradicated.
Racoons have also bred, but never very successfully.
Mongolian Gerbils have bread in the wild in the UK.
Hialyan Porcupine has bred in the wild in the UK, in Devon in the 1970s but were eradicated before getting established.
Père David's Deer have also bred in Northamptonshire 197's and still believed to be at large.
An excellent read and reference is The Naturalized Animals of Britain and Ireland, by Christopher Lever. and covers mammals, birds, reptiles, amphibians and fish!!!