Somebody’s been watching their Incredibles DVD.
<pant, pant> I got here as quick as I could! <pant>
Perhaps. In fact, I think likely.
Standing (or even easier, lying) on water due to surface tension alone is certainly possible in theory, if gravity is low enough (and you’re nicely waxed up to get maximum repulsion). After all, there is a definite surface tension force, and so you could always reduce gravity until your weight is lower than that force.
But in practice, any tiny movement you make could be enough to destroy the surface tension, and then down you go (though more slowly, given the lower gravity).
On the other hand, if gravity is very low, it might well be possible to generate enough force to keep you upright, not from surface tension, but from actively pushing water with your feet (I believe this is what the running lizard does). Kind of swimming, just in an upright position, with most of your body out of the water. Questions about whether it’s humanly possible to balance, keep upright, etc. remain, but there’s no physics reason why some creature the size of humans couldn’t run on water, given sufficiently low gravity.
It’s been done at least once in human history. 
And at least once in dog history. The first time my old dog Bear (rest his soul) encountered a pond, we pushed him off of a raft in the middle (about 20 feet from the shore). He doggy-paddled so hard that only his feet were even in the water. Most of his back legs and about half of his front legs were wet, but not his belley or chest. He never did tell how he managed it, though.
If I may add my two cents, I think that it is 100% possible for a human to even stand completely stationary on water and here is why: The speed of displacement is relative to amount of space you have to displace, i.e. if you had a foot shaped container of water and put your foot on it and stood the water will displace out of the container at a slower rate then in a lake. But then I guess I am taking this out of a real life situation I mean are there many size 12 mens lakes around that I can run across?
Less anecdotally, it’s done all the time in the colder areas of the planet (Speed required: 0 kph, and yes, it’s easier in low gravity), but I doubt either is what the OP was looking for.
Any one seen this video of running on water?
I think it’s well accepted that certain zombies can walk on water.