Are you pinging from the Management interface or TMM/data interface? If from the Management interface, it's unlikely to be a BigIP problem. Nevertheless, as a first step, try to set a MAC masquerade address for your BigIP traffic-group. This is normally only needed for Active-Active BigIP clusters but it doesn't hurt with Active-Standby. A problem with
Gratuitous ARP
updates and old ARP caches is a possibility. You may need to perform a similar procedure with your L3 switches to make the failover event more reliable.
SOL article
Video explanation by J.Rahm