Yes you can foul on a push by pushing
https://youtu.be/CPGXl8QAUSI?t=4669
Never seen that before.
Never even heard of anyone calling it.
Occurred at a World championships match.
The angle is hard to see for me, but it does not look like it was a push shot, yes he shot it soft but it looked like it went through the cueball smoothly without any secondary or too long of a contact.
I have seen players do nothing but "call" a push out and not actually touch the cueball, just say "push" and hand over the table to the person. Which is not legal.