You've got a layer of harder rock there, where the water is perking out. That, or a seam of fire clay. Depends on if you have coal in the area.
It'd be handy to have a hoe, and dig in that hole you have, to see just how far down it is to the rock, or clay. The trick is, to go up the slope far enough to catch it, before it has a chance to perk out. Not right there... No less than 36" down to the rock, or clay. It's also nice to know if the layer of material the water is coming off of, is pretty flat, or turned up a bit there, where it's coming out now. A level and rod, or laser should tell you that pretty quick. That will tell you whether to cut straight across the driveway, or allow you to go at a slight angle, but not much to your outlet side.
If it's flat, you can angle. If the material goes deeper, as it goes back into the hill, better come straight across. If the rock is deeper, than where it comes out now, and you dig a trench, and fill with gravel, unless it is able to get away from where your starting point is, all you will be doing is making a collection point. And that water will more then likely soak it's way back out to where it is now. A slight fracture in the rock will let a lot of water flow over a period of time. A collection of water 6 to 8" deep, seeping out through cracks, just gives it more head, and volume.
If it is fire clay, you'll more then likely hit a thin seam of coal, or "blossom coal" as we call it. If this is the case, do NOT dig through that fire clay. At this point, it is your friend. Water has not gone through it, since it was formed, and it a great water proof bottom for a trench. Unless it has counter rotated towards the outside edge of the hill, it should taper downhill. I've seen fire clay as thin as 1/2", so you'll have to be careful, if that is what it is...
A 12" wide trench would probably suffice to catch that, but use your own judgement. I would spring for the 4", double walled perforated plastic pipe. It generally comes in 20' sticks. We used #8 river gravel, or pea gravel. Just make sure to fill as high, as you see water perking out of the upper side of your trench. As mentioned, some geotech, or something similar to keep dirt from washing down through the stone. For my own personal use, I used the higher grade weed barrier cloth. A good sharp box knife will cut the barrier material, and give you double length of the roll. Heck, we used to use straw about a foot thick loose, before the geotech stuff.
If and where the rock drops off beyond that point, it may be an idea to go to solid pipe, to carry it away, ending the stone at the perforated/solid joint. We've found using perforated, and stone beyond the hard surface the water is coming off of, allows it to soak out, and seek the path of least resistance, and have the same problem somewhere on down the line. Just pack some good fine clay around the joint to keep the water in the pipe. Some of the material on, or around the perforated pipe back a few feet from the joint, is a good idea too.
And don't forget a varmint barrier for the end of the outlet. Muskrats, and I'm sure other critters love that stuff... Maybe because some is made from soy based materials...
We always slipped a piece of galvanized perforated pipe over the end, to protect it. But, again that's up to you.
