More just a general interest here than a specific need but ....
In a previous job in London we had dozens of Aussies and S. Africans over on three year visas - they could work but had to return after it expired.
So, if someone was here on a short-term working visa, can a business sponsor them to stay on after it expires?
Yes it can happen but it does depend on cirumstances - i.e skill shortage and the job that they will be doing.
I was sponsored in Australia after my working visa expired, I did return though but the option is there in the future if I wish to get sponsored again.