As others have already stated, in general pings are ICMP-based and have no ports. There is, however, such a thing as TCP Ping where, instead of the typical 3-way TCP handshake, only the first 2 steps are performed and the delay between is measured. Once the measurement has completed, a RST ACK is sent to close the half-open connection. Then the process repeats until the counter/duration is reached or you terminated the process. Using TCP Ping (which I use FREQUENTLY to test for open ports on servers my systems admins work on) you are able to specify destination ports to test (to verify a server is listening on a certain port). The source port is just an ephemeral random port.
If you'd like to see an example of a TCP Ping utility (the one I use on Windows systems), here you go:
TCPing. Also, NMAP comes with a utility called NPING which has a flag to allow it to perform TCP based pings too.
As a note, some network equipment also has this capability, such as Cisco ASAs using some of the newer operating system versions. The command is:
ping tcp <destination IP> <destination port>