Is it possible to register an ".er" domain?
No. It is not available for registration at this time. More
No, it is not possible to register .er right now (Dec. 2016).
One way to check how difficult is to register within a domain extension is to perform a search in google like "site:.er". For .er, you can see that google has detected only 10 domain names, and most of the are from the government.