A linguistic isolate is a language that can't be shown by comparative methods to be related to any other language spoken at present. The best known example of an LI is Basque which is spoken in Spain and the Pyrenees. I have no information that Basque is presently endangered, since Basque people are very proud of their linguistic heritage and are militant in their struggle. Other LIs are Burushaski, Okinawan, Ainu(extinct?), Sumerian and Pictish(both extinct). There is some evidence that Japanese and Korean are related because of similar syntax structure. For this reason I would not include them on the list of LIs. Feel free to add to the list.

Okinawan is universally accepted as the closest living relative of Japanese. The family to which both of these belong to is known as "Japonic".

Ainu is not extinct, but there's evidence that it is also related to Japonic and Korean within a larger Altaic family.

The genetic affiliations of Pictish have not been demonstrated to everyone's satisfaction, but there's a strong case for it being an Indo-European language of the Celtic family.

Right you are. I forgot about the Ryukyu languages. However, I would add Tarascan and extinct Etruscan to the list. I'm wondering how much evidence is left of Pictish. Were Picts literate in their own language? Without surviving parchments I don't see how any conclusive judgement can be made at all.

Some of the Pictish royal names include Celtic-style patronymics, so I'd assume that the language was Celtic.