Quietly speak in support of Argentinian lady and disappear (9)
I believe the answer is:
evaporate
'disappear' is the definition.
(I've seen this before)
'quietly speak in support of argentinian lady' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'speak' becomes 'orate' (orating is a kind of speaking).
'in support of' says to put letters next to each other (I've seen this in other clues).
'argentinian lady' becomes 'eva' (Eva Peron, wife of Argentine president Juan Peron).
'p'+'orate'='porate'
'porate' after 'eva' is 'EVAPORATE'.
'and' is the link.
(Other definitions for evaporate that I've seen before include "Disappear into thin air" , "Dissolve into the air" , "Turn from liquid to vapour" , "Change from liquid to vapour" , "Go from liquid to vapour" .)