Single quotes prevent variable interpolation. Get rid of them around $file everywhere:

#!/usr/bin/perl -w
use warnings;
use strict;

use Archive::Zip qw( :ERROR_CODES );

my @files;
@files = <ATP.xxxxxx*.zip>;
foreach my $file (@files){
 my $zip = Archive::Zip->new();
 $zip->read($file) == AZ_OK or die "read error $!";
 $file=~ s{\.[^.]+$}{};
 my $m1 = $zip->memberNamed($file);
 $file=~s/^.{4}//s;
 $m1->fileName($file);
 $zip->overwriteAs("$file.zip") == AZ_OK or die "write error $!\n"
}
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