How can I check the extension of a file using Perl?

You can use File::Basename for this.

#!/usr/bin/perl

use 5.010;
use strict;
use warnings;

use File::Basename;

my @exts = qw(.txt .zip);

while (my $file = <DATA>) {
  chomp $file;
  my ($name, $dir, $ext) = fileparse($file, @exts);

  given ($ext) {
    when ('.txt') {
      say "$file is a text file";
    }
    when ('.zip') {
      say "$file is a zip file";
    }
    default {
      say "$file is an unknown file type";
    }
  }
}

__DATA__
file.txt
file.zip
file.pl

Running this gives:

$ ./files 
file.txt is a text file
file.zip is a zip file
file.pl is an unknown file type

Another solution is to make use of File::Type which determines the type of binary file.

use strict;
use warnings;

use File::Type;

my $file      = '/path/to/file.ext';
my $ft        = File::Type->new();
my $file_type = $ft->mime_type($file);

if ( $file_type eq 'application/octet-stream' ) {
    # possibly a text file
}
elsif ( $file_type eq 'application/zip' ) {
    # file is a zip archive
}

This way, you do not have to deal with missing/wrong extensions.

How about checking the end of the filename?

if ($file =~ /\.zip$/i) {

and then:

use strict;
use Archive::Extract;

if ($file =~ /\.zip$/i) {
    my $ae = Archive::Extract->new(archive => $file);
    my $ok = $ae->extract();
    my $files = $ae->files();
}

more information here.