I had already overwritten the offending code with a procedural version which is why I didn't provide it. I have however re-produced the problem below with a much less complex compare routine. I generated a file of 1 .. 100_000_000. I then split that into ten 10_000_000 line files. Then I ran perl merge.pl > source.merged and watched the memory usage continue to climb.

#!/usr/bin/perl
use strict;
use warnings;
use List::Util 'first';

my $finish = 'The end is near';

my @list;
for (glob('source_part_??')) {
    open(my $fh, '<', $_) or die "Unable to open '$_' for reading: $!"
+;
    push @list, $fh;
}

my $fetch = sub {
    my ($fh) = @_;
    return $finish if eof $fh;
    return scalar <$fh>;
};

my $compare = sub {$_[0] <=> $_[1]};

my $next = gen_merger(\@list, $fetch, $compare, $finish);
while (1) {
    my $item = $next->();
    last if defined $item && $item eq $finish;
    print "$item\n";
}

sub gen_merger {
    my ($list, $fetch, $compare, $finish) = @_;
    my @item = map $fetch->($_), @$list;

    my $done;
    return sub {
        return $finish if $done;
        my $idx = first {$item[$_] ne $finish} 0 .. $#item;
        my $next = $item[$idx];
        for ($idx + 1 .. $#item) {
            next if $item[$_] eq $finish;
            my $result = $compare->($next, $item[$_]);
            ($idx, $next) = ($_, $item[$_]) if $result == 1;
        }
        $item[$idx] = $fetch->($list->[$idx]);
        $done = 1 if ! defined first {$item[$_] ne $finish} 0 .. $#ite
+m;
        return $next;
    };
}
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Note: This version has one more bug fix in the line immediately preceding return $next. The range was changed from $idx .. $#item to 0 .. $#item.