Can a Paladin have more than one Smite working at the same time against different foes?
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It's not stated that they can't in the core rules, so I would say that you can if your level is high enough to warrant multiple uses of Smite per day.
In my opinion it kind of goes against the intent of Smiting, but there seems to be no rules reason why you can't. Much the same way you can smite a target and then switch to attack another target to get an AC boost. If Paizo didn't want players to be able to do this they would have specified.