From the text of the directive:
"Natural persons may be associated with online identifiers provided by their devices, applications, tools and protocols, such as internet protocol addresses, cookie identifiers or other identifiers such as radio frequency identification tags. This may leave traces which, in particular when combined with unique identifiers and other information received by the servers, may be used to create profiles of the natural persons and identify them.
An IP address is an "identifier". However, an IP address does not in and of itself identify a natural person; you know that, I know that, and even the GDPR knows that.
However, if you start building a map of IP addresses to user real names, or some other form of profile construction, then the IP addresses become personal information.
(comment hoisted from other thread)