Self-obsessed Dutchman on mountain has style (9)
I believe the answer is:
rembrandt
'self-obsessed dutchman' is the definition.
I can't judge whether this defines the answer.
'on mountain has style' is the wordplay.
'on' becomes 're' ('on' can be mean 'with reference to').
'mountain' becomes 'm' (geographical abbreviation).
'has style' becomes 'brandt' (I can't justify this - if you can you should believe this answer much more).
're'+'m'+'brandt'='REMBRANDT'
Can you help me to learn more?
(Other definitions for rembrandt that I've seen before include "'Night Watch' painter" , "Dutch painter, 1606-69" , "Influential Dutch artist" , "Dutch painter, d. 1669" , "One notably exhibited" .)