In a while (<>) loop, "eof" or "eof(ARGV)" can be used to detect the end of each file (while "eof()" will only detect the end of the last file). I'd try:

my $binmode = 1;

while (<>) {
   binmode ARGV, ":utf8" if $binmode;
   $binmode = eof;   # Don't add parens!

   do_whatever($_);
}
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