Apparently, the ARP process within the ESP8266 somehow "dies" after a while. With the new AT+PING one can at first see the PING-request frame to the target-IP, then the target issues an ARP broadcast to find out the MAC address of the ESP-station, but that ARP request never gets answered by the ESP. Thus no PING-reply…. Same is basically true for all other IP communication.
As a workaround, I then made a permanent entry to the target PC’s ARP table (on Windows use “arp -?” in a command window, you need to be in admin mode to make a static table entry), so there is no need for an ARP address resolution. From then on, the communication did not fail again, regardless if it was from AT+PING or to the device’s TCP-server (AT+CIPSERVER). (My ESP is at 0.9.5 and is run in station mode)
As suggested, I’ve also tried the 0.9.6 b1 beta, but this versions seems to have still the same issues. Besides, it crashes from time to time. So I went back to 0.9.5 which turned out to be rather stable – as far as I can tell by now…
JoeK