It sounds like the capabilities on iputils-ping could be messed up. Ping has always needed special permissions or attributes set to be usable by nonroot, and iputils uses capabilities. So you should check them (the proper ones seem to be cap_net_raw+ep)
# getcap /usr/bin/iputils-ping
/usr/bin/iputils-ping = cap_net_raw+ep
That should also be the output for iputils-ping6...
If it's not, then you should try to set them:
# setcap cap_net_raw+ep /usr/bin/iputils-ping