Well, as far as I can tell, you ask the user about the file _outside_ of the while-loop, and you don't take into consideration that $dir might be something else than '.'
This is how I would have written it (more or less):

my $dir = shift || '.';

opendir DIR, $dir or die "Can't open directory $dir: $!\n";
while (defined (my $file = readdir DIR) ) {
      next if $file !~ /^\w{10,}\d{4,}\.\d{1,}/;
      print "Found a file: $file\n";

      print "are you sure that you want to delete $file   'Y/y(es)'  "
+;
      chomp( my $answer=<STDIN> );
      do { unlink "$dir/$file" or die "Can't delete $file: $!\n" }
         if $answer =~ m%^Y(es)?$%i;
}
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You might have to use the 'while( defined (...) )'-construct if you are using an older perl.

Autark.