Banned

I have gone into a customer account and changed their status to
"Banned - Not allowed to login or shop".

Yet this customer keeps placing orders. Each time he uses the same email address.

How does this feature "Ban" the user, by IP address or Email address?

If this feature does not work, I guess the next best thing is to delete his account.

Deleting the account just lets the customer make a new one ...

Look in the Customers ... Customers ... and see what the Customer is set to for the Customer Authorization Status ...

Then check that the file:
/includes/modules/pages/login/header_php.php

is testing for the customers' access status, there should be a test where you see:

      // this account is banned

Customers --> Customers - There is a big RED "X" in the Authorized column for this user.
Clicking on Edit
Customers Authorization Status "Banned - Not allowed to login or shop"

/includes/modules/pages/login/header_php.php:

// this account is banned
  $zco_notifier->notify('NOTIFY_LOGIN_BANNED');
  $messageStack->add('login', TEXT_LOGIN_BANNED);
} else {

Using Version 1.3.9e

Can you make a test account and ban yourself and see if it works for you?

By chance do you know the users IP address?