I see a couple of problems. Look here:

foreach $line (@lines) {
    my $id = $line =~ m/ID No: (\d+)/;
    if ($seen{$id}++){
# ...
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You're going to get 1 for $id all the time because the match is in a scalar context instead of list context.

The other thing is you want to say "if ( ! $seen{$id}++ ) ...". The first time an $id is seen, $seen{$id}++ will be false. Every time after that, it will be true. Since you want it to be true once and false ever after, add the negation.

Here's the part I copied, rewritten:

foreach $line (@lines) {
    my ($id) = $line =~ m/ID No: (\d+)/;
    if ( ! $seen{$id}++){
# ...
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I some other suggestions for you.

First and most important, check the value of open! You try to open your files, but if they fail, you'll never know. Also, it's a generally good idea to use lexical filehandles instead of the global ones you're using.

open my $in_fh, '<:utf8', 'input.txt'
    or die "Can't read 'input.txt': $!";
my @lines = <$in_fh>;
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Second, you're doing everything in memory. If your input is huge, you could run out of memory. Instead of reading every line and then looping over them, consider using a loop that reads a line at a time instead. You'd also want to open your output files at the start and write into them during processing instead of collecting their eventual contents in (memory-based) arrays. Like this:

open my $in_fh, '<:utf8', 'input.txt'
    or die "Can't read 'input.txt': $!";
open my $purge_fh, '>>:utf8', 'purge.txt'
    or die "Can't append to 'purge.txt': $!";
open my $uniq_fh, '>:utf8', 'data.txt'
    or die "Can't write 'data.txt': $!";

my %seen;

while ( my $line = <$in_fh> ) {
    my ($id) = ($line =~ /ID No: (\d+)/);
    if ( ! $seen{ $id }++ ) {
        print $uniq_fh, $line;
        $new_uniq++;
    }
    else {
        print $purge_fh, $line;
    }
}

close $in_fh       or die "Close failed for input: $!";
close $purge_fh    or die "Close failed for purge.txt: $!";
close $uniq_fh     or die "Close failed for data.txt: $!";
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Finally, if you don't already, Use strict and warnings!