Llívia                  (Spanish exclave in France)

A 12,84 km² territory of the Spanish Pirineus region. Population about 1200, east-southeast of Andorra.

The northern area on the map is a property of Llivia belonging to the Republic of France.

Llívia is a town of the region of Cerdanya (that's the name in Catalonian; it is Cerdaña in Spanish and Cerdagne in French). This historical region is divided between the French and Spanish states. For the Catalonian nationalists, all of Cerdanya belongs to the Països Catalans (Catalolian countries); so, in that case, Llívia wouldn't be an exclave. 

Llívia official site www.llivia.com defined the territory as an «enclavament de l’Estat espanyol en l’Alta Cerdanya»; that is: an «enclave of the Spanish State in High Cerdanya». Lower Cerdanya is the territory south of the French-Spanish border.
The Catalonian nationalists call the Catalonian-speaking French region north of the border as Catalunya Nord (Northern Catalonia).
The official languages in Llívia, as in the rest of Catalonia, are Catalonian and Castilian (better known abroad as Spanish.
Catalonia (Catalunya in Catalonian; Cataluña in Spanish; Catalogne in French) is an autonomous region inside the Kingdom of Spain. It's similar to Greenland or Faroes Is. in Danmark, or even more autonomous. Catalonia has its own parliament, government, police force, flag, national anthem, national day, language, media, etc. The Catalonians are very proud of their history, culture and language. The regional ("regional" in the Spanish State, de jure definition) government of Catalonia is controlled by a moderate nationalist party that promotes Catalonia as a non-Spanish nation that history led to be under Castilian (=Spanish) domination for some centuries. Nowadays, with the autonomic statute, they are short of be an independent nation. Indeed, some de jure independent nations have less self-government than Catalonia.
The Catalonian language is also spoken in Balearic Islands, Valencia Region, Franja de Ponente (Eastern stripe of Aragón Region), Alghero city (in Sardegna, Italy), South-east France / Roussillon (Catalunya Nord), and Andorra (the only independent nation where it is an official language). The extreme nationalist Catalonians see all those territories as the Catalonian Nation, and they call it Païses Catalans. Those extreme nationalists are especially active in Catalonia, Balearic Islands and Valencia.

Acknowledgement: Mr. Francisco José Nunes dos Santos

Links:
Eef Bern's The bordermarkers of Llivia