Can allied pings from specific players be prevented?

Yes, it is now possible to mute ally pings (as of 19 April 2016) by clicking the mute player pings button on the in-game score screen (opened with Tab).

No, they cannot be prevented (as of 13 April 2015).

As the game in its current Beta state, you can only block communication to a player in your team from the game statistics menu in or after game. That feature you asked for is not implemented yet. You can always give your feedback about the matter in official forums.

For workaround, if you currently navigate to options menu and then disable all sound for muting pings, then you can use the keyboard shortcut for toggling the sound: Ctrl+S