Hmmm ... well you could try Fcntl; however, it may not be supported on your platform:

!#/usr/bin/perl

use Fcntl;

open( FH, "<", "/bin/cat" )     or die "cannot open file: $!\n";
$flags = fcntl(FH, F_GETFL, 0) or die "Can't get flags for the file: $
+!\n";

if( $flags | O_BINARY ) {
   print "OPEN FOR BINARY\n"
}
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