No. "Vie" is a feminine word, so that's "la vie". "La vie chic" could be grammatically correct, in such expressions as 'elle voulait mener la vie chic des parisiennes' (she wanted to enjoy the chic life of the Parisians', but chic is barely used as an adjective, except in short, informal expressions: c'est chic, tu es chic (you're classy), etc...
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