An ETA is simply an authority for a visa exempt passport holder to board a plane for Canada, it is not a visa or authority to enter or reside in Canada. Do you really mean a TRV ?
Others can correct me but your wife is not residing/living in Canada currently she just regarded as a visitor Canada with a limited stay so her country of residence is where she has permanent residence which is not Canada. If she were in Canada as a student or on a work permit the answer could be different given techincally she would be residing in Canada, but just as a visitor not.
A TRV by the way as sure you know on its own is only a travel document authorising the holder to travel to Canada and does not guarantee the holder entry to the country or permission to stay for any specific period that would be determined by CBSA at the border. So whilst a TRV has a validity date it is not any indication how long anyone can visit or reside in Canada.
Having said above guess no harm in including a copy of a TRV, a copy of an actual ETA would serve no value.