Oh yes, definitely! I'm afraid I don't know the details for the second part of the question, but exile was most definitely a form of punishment throughout European history. The earliest concrete example that springs to mind is ancient Athens, whose practice of voting malcontents and public pariahs out of the city for a decade gives us the word "ostracization". The latest example in my head is Napoleon, who was (twice) exiled, because the monarchs of Europe considered execution an unfitting end for even a self-made Emperor.
Exile was formerly a very potent form of punishment, for two primary reasons. First, emigration was very uncommon in the pre-modern period, and few areas greeted newcomers and outsiders with anything other than hostility. Moreover, a person's economic, social, and political status were tied to the people, resources, and community of their hometown, and none of these could travel from place to place easily. Only someone with enough wealth or influence in their new home to purchase economic security could succeed comfortably in exile, and even then much was lost in the transition.
Re the first question, yes: here's a post on Iceland
I suppose an interesting perspective could be wrought if someone could discuss the history of asylum law.
Banishment still occurs in the United States, btw. Paroled sex offenders, for example, are often limited with respect to their movement. There have been various legal challenges but the laws tend to stand.
The British government actually transported a lot of convicts to her colonies. First to the Americas and after losing those colonies to Australian penal colonies in the 18th and 19th century. Over 165.000 convicts were exiled to Australia over the years. According to this entry it was punishment for crimes not severe enough for capital punishment, for example this one.
Two examples I can give you: one from Euripides's The Medea and one from a more modern case, though it wasn't truly a government that did so.
In The Medea (a continuation of the stories of the mythical figure Jason, as in Jason and the Argonauts, golden fleece, etc.), Jason has decided to marry the daughter of the King of Corinth, Creon, and forsake his relationship with Medea. Worried of what she might do upon hearing this news, Creon orders that Medea be banished from the city after a period of one day.
The other, actual instance of banishment being used comes during the time of the Protestant reformation in Europe in the 1530s. A group of protestants called the Anabaptists ("to baptize again") lead by Jan Matthys (several spellings exist) took over the town of Münster and declared it their New Jerusalem. During the beginning of their control over the town, the leadership of the group including Matthys stated that either the residents rebaptized into Anabaptism or were to be immediately exiled from the city. These kinds of sentences of exile were basically sentences to ruin: those exiled could only leave with the clothes on their back into the harsh winter. Seeing as everything of value a person might have in that time was contained within their home (including their home) and these people were permitted to take none of it, those who managed to find a life in another city did so with absolute desperation and with no amount of wealth at all.
Not a historian, so I can only share a very general piece of trivia, but our modern word "ostracism" comes from the ancient Athenian practice of democratically voting for an individual to be exiled for ten years!
It wasn't a punishment for crimes, though. The people could vote to hold an ostracism once per year, and when it was held, eligible citizens could write down any name they liked - even if the person had done nothing wrong. If the person with the most nominations met the minimum required number of nominations, he'd be exiled for ten years.
EDIT: someone already mentioned ostracism below - sorry!
