Yay! With the code below it worked:

#!/usr/bin/perl
use Crypt::SMIME;

open(PKEY, "/home/alice/ssltestkey.pem") || die "open failed: $!";
open(CERT, "/home/alice/ssltestcert.pem") || die "open failed: $!";

my $plain = <<'EOF';
From: alice@example.org
To: bob@example.org
Subject: Crypt::SMIME test

his is a test mail. Please ignore...
EOF

my $privkey = do { local $/; <PKEY> };
my $crt = do { local $/; <CERT>};

my $smime = Crypt::SMIME->new();
$smime->setPrivateKey($privkey, $crt);
my $signed = $smime->sign($plain);
print $signed;
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Thank you andal, Corion, sundialsvc4 for your suggestions!