Without hacking into another system on the LAN and routing your packets that, it is not possible.
That said, depending on the network there are things you can do to make the data less trackable, including, if possible, changing the IP address you use (ie ignoring DHCP and setting up something else). This is not a good idea if you don't know the network topology because you may conflict with someone elses address and break things.
Another thing you may be able to do is to modify the MAC address associated with your network/wifi card. This will tell people the incorrect information about the device. If you change it regularly it will be difficult to pinpoint the device based on the source IP/MAC - although that still leaves them with the ability to track based on destination, time-of-day or other such parameters.
Another thing you could possibly look towards, depending on your goals (I know it has been done, I've never attempted it so I don't know the software used), could be to set up a hidden connection using steganography - foe example using a VOIP conversation as a kind of VPN wherein you imbed the information you want - this is very inefficient though).