Real-world examples of Darwinian extinction
Although you don't mention it in your question, I guess you'd like examples of extinction in (more or less) constant environment? Then I guess you'd like cases of extinction due to the introduction of an overseas species out-competing the original one. 

Red squirrel is not extinct yet, but its sharp population decrease is mostly linked with competition with newly introduced grey squirrel, although habitat loss and concurrent introduction of a virus are also part of the causes. Read at https://en.wikipedia.org/wiki/Red_squirrel