No, actually quite a few countries don't have postal codes.
"In February 2005, 117 of the 190 countries members of the Universal
Postal Union had post code systems. Examples of countries that do not
are: Ireland (although a national postal code system will be
introduced in 2008 ), Hong Kong, and Panama."
List according to UPS of countries that don't have zip or postal codes.
Does Ireland use postal codes? What form do Irish postal codes take?
"In Ireland, only Dublin City has postal codes, and they are really
postal districts. You will notice that all of Dublin City centre
establishments are in "Dublin 1" or "Dublin 2", etc. Outside the city,
it is simply County Dublin. The postal code system for Dublin is
relatively simple. Odd numbers are addresses on the North side of the
River Liffey. Even numbers are on the south side (except for parts of
Dublin 8, which encompasses part of the Phoenix Park). Dublin 1 is the
North City Centre. Dublin 2 is the South City Centre."
FRANK'S COMPULSIVE GUIDE TO POSTAL ADDRESSES
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Google Search Terms Used: which countries do not have postal codes