Extraordinary sense finally awoken in one inspiring language (9)
I believe the answer is:
esperanto
'language' is the definition.
(Esperanto is an example)
'extraordinary sense finally awoken in one inspiring' is the wordplay.
'extraordinary sense' becomes 'esp' (extra-sensory perception).
'finally' says to take the final letters.
'in' means one lot of letters goes inside another.
'one inspiring' becomes 'erato' (muse or inspiration in Greek myth).
The final letter of 'awoken' is 'n'.
'n' inserted inside 'erato' is 'eranto'.
'esp'+'eranto'='ESPERANTO'
(Other definitions for esperanto that I've seen before include "Universal language" , "Dr Zamenhof's language" , "Artificial international language devised in 1887" , "Parse note in the artificial language" , "Modern language" .)