Split here in jacket across arm’s lining, cracking up? (7)
I believe the answer is:
croatia
'split here in jacket across arm's' is the definition.
The answer is a location as well as being a singular noun. This is suggested by the definition.
'lining cracking up?' is the wordplay.
'lining' becomes 'croat' (I can't justify this - if you can you should believe this answer much more).
'cracking' becomes 'AI' (resembles Ac, informal term for excellent).
'up?' is a reversal indicator (in down clue: letters go upwards).
'ai' reversed gives 'ia'.
'croat'+'ia'='CROATIA'
Can you help me to learn more?
(Other definitions for croatia that I've seen before include "From small country" , "European state" , "Land in South-East Europe" , "Balkan republic, capital Zagreb" , "Adriatic country" .)