Shouldn't that be  map { "\"$_\"" } (added missing underscore), or perhaps better  map { qq{"$_"} } or even just  map qq{"$_"}, (note: comma is needed)?

c:\@Work\Perl>perl -wMstrict -le
"my $s = join '^', map qq{\"$_\"}, 'foo bar', 'baz boff';
 print qq{:$s:};
"
:"foo bar"^"baz boff":
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