Is Bergania a country?
No-- it is a fictional country made up by Eva Ibbotson in her
novel The Dragonfly Pool. Many of the events in the story don't
match with the facts of history, and this would have been more
jarring if she had used a real country.
However, the country of Bergania as described in the book bears
significant similarity to the real country of Liechtenstein. Like
Bergania, Liechtenstein is a small country located between Germany
and Switzerland. Like Bergania, Liechtenstein was neutral during
World War II, and accepted hundreds Jewish refugees from
Germany.
There are other similarities, too. I enjoyed reading the
Wikipedia entry on Liechtenstein and then looking for similarities
as I read the book. There are many.