How can I set up a date picker so the day will automatically be blocked off when someone else picks it?

I'm using a Date Picker. I would love it if when the user picked a day, that day would automatically be blocked off (greyed out) so  no one else can pick it.

Can Date Picker be set up like that? If so, how?

That is not possible to achieve with the Date Picker widget.

If you want to disable selected dates, you can use the Date Reservation widget. 

Thanks, but...
  You CAN block days, even stretches of time, using the Date Picker
widget. I just can't figure out how to have it happen automatically (ie,
when the client clicks on a date, it greys out).
   I'll review your information below.
Regards,
Carol
...

Hello Carol - You're most likely referring to the DATE PICKER FIELD (and not the DATE PICKER WIDGET). Yes, you can limit (but not block) dates within the date picker field but this won't prevent other users from picking that same date again.

The DATE RESERVATION WIDGET suggested by my colleague works like a regular reservation/booking. Once someone picks a date, then that date can't be selected by anyone else anymore.