I have an old raspberry pi 2011.12. It has installed an up to date raspbian (headless).
It is connected to my local wifi network and internet. SSH is enabled. The following commands work fine on the raspberry:
ping google.com ping loclahost ssh localhost
However when I ping to raspberry from other devices (tried few of them) or ssh - it does not work. I am trying to access using ip address 192.168.1.167. The ip address I determined correctly (it is different for wifi and ether connection). It is working fine when I connect the raspberry to my router via cable.
P.S. It may be the issue not only with wifi connection but it seems with ethernet I have the same issue. Sometimes this issue disappears but in most case I cannot ping the board from outside.