The recent upgrade of dnsmasq to version 2.90 fixing a security issue is known to break the system for multiple users. The reason for this breakage is currently investigated, but yet to be found.
A workaround is applied to postmarketos-base-ui-networkmanager However, your system might prefer a mobile connection over WiFi until this situation is properly resolved.
Recovery
If you already upgraded to the new version and need to recover from there, a solution would be to create a config file that deactivates the DNS cache:
$ cat /etc/NetworkManager/conf.d/51-disable-dnsmasq.conf
[main]
dns=default
After a restart, you can see DNS resolution working again. This is a workaround, and meant to be temporary.
Now, you can run sudo apk upgrade -a, to get the new version of postmarketos-base-ui-networkmanager and delete the config file /etc/NetworkManager/conf.d/51-disable-dnsmasq.conf again.
Related: pmaports#2601, aports#15788, pmaports!4824