To get indefinite depth, you'll need a recursive solution

Not quite. You simply need a stack. Recursion is just one way of obtaining a stack. What follows is another:

my @to_visit = $base_dir;

while (@to_visit) {
   my $dir = pop(@to_visit);

   opendir(my $dh, $dir) or ...;

   my $file;
   while(defined($file = readdir($dh))) {
      next if $file eq '.';
      next if $file eq '..';

      # Should use File::Spec.
      $file = "$dir/$file";

      if (-d $file) {
         push(@to_visit, $file);
      } else {
         ...
      }
   }
}
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Substitute pop with shift if you want to process the tree breadth first.