In river, Borodin's prince gets a sense of chilliness (5)
I believe the answer is:
rigor
'chilliness' is the definition.
(thesaurus)
'in river borodin's prince gets a sense' is the wordplay.
I cannot really see how this works, but
'river' could be 'r' (abbreviation for river) and 'r' is found in the answer.
'prince' could be 'igor' and 'igor' is present in the answer.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).
'of' acts as a link.
Can you help me to learn more?
(Other definitions for rigor that I've seen before include "showing failure to respond" , "Stiffness (after death?)" , "Shivering chill before fever" , "- - mortis" , "Chilled shivering often a symptom of disease" .)