mixture, the statement 'zip codes do not have spaces' is correct. They don't. The zip code is the US postal service's invention. The rest of us use things called post code (UK), postal code (Canada), PIN code (India), code postal (France), PostLeitZahl (Germany), etc. etc., which may or may not have spaces.
As already said, the UK post code is usually written with a space for ease of reading, but the space isn't required by the Post Office's systems, which can parse a valid post code with or without a space..