Even if this may not help you in this particular issue, I suggest running with strict and warnings enabled; warnings tell that you'll probably want to substitute:

   print STDOUT @NewRow[0], " ", @NewRow[1], "\n";
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with

   print STDOUT $NewRow[0], " ", $NewRow[1], "\n";
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to actually print the first two elements in @NewRow. I would also avoid using $Record:

open( BEST, "< $best_file") || die "Sorry, can't open $best_file: $!";

my @best_array = ();  # Silences strict

while( <BEST> ) {
   print STDOUT "\t Reading next line...\n";
   chomp;                    # Works on $_
   s/^\s+//;                 # Ditto
   my @NewRow = split /\s+/; # Ditto, note scoping with my
   print STDOUT $NewRow[0], " ", $NewRow[1], "\n";
   push( @best_array, \@NewRow ); # Take reference, avoid copy
   print STDOUT "\t Proceeding to next line...\n";
}
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Other than that, I tried your piece of code and runs smoothly in v. 5.8.5 in cygwin.

Update: removed annoying hanging sentence