I kind of had the same problem these days. And the same culprit could be causing your problem, too.
The culprit is the alternator.
Here goes the story... I recently had to replace my ECU and I did replaced it with a sh virginized one, due to Error code P0351. And as a possible fault for that, i discovered that the coil no. 1 was faulty, having short cut inside. So that could have toasted the ECU, anyway, toasted it was and I replaced it.
Couple of days ago, having the alternator whistling, I got it under my radar. And I ran into this very interesting information, a simple test that we can do to know if the alternator's diodes bridged works correctly. The test is: using the multimeter, to measure the AC voltage that comes out of the alternator. And it should be under 50 mili Volts. And I love this idea, because it makes a lot of sense. The "raw" current produced by the alternator, by spinning of the rotor inside the stator, is AC. And the AC is turned into DC by the diodes bridge. And if some diodes are faulty, there is AC being put out by the alternator. So I got my multimeter and I got a shocking value of 30.3 V AC out of alternator.
So, I bought a new diodes bridge and put it into my alternator. Next measurement, 30.2 V AC out of alternator. It turns out that the multimeter I used is a simple one and cannot read AC voltage in a DC system. The real AC voltage that the alternator gives now, with the new diodes bridge, is 0.1 mili Volts, so this bridge works perfect.
But, inside of the alternator, I discovered another problem. One of the slip rings was used, 1mm more than the other, and in the same time it was covered in a dust film which I'm pretty sure, was preventing a good continuous contact between slip ring and brush. Thus, the result, alternator not putting out enough voltage. The multimeter was reading between good range 13.8 - 14.2 V DC, but I'm sure that if I've put an oscilloscope onto it, it would have read big droppings of that voltage, not continuous voltage. Therefore, the battery got to be flat.
I went with the battery to the local battery service center, the battery is 4 years old, out of warranty, but they do free mesurments and charging of battery. So they are really nice. The stored quantity A, that is needed for starting up, was 50 in my battery, normal value for full battery is 480 or 500 A (can't remember exactly which, but there). The guy said that I cannot crank the engine with that battery and he couldn't belive it when he saw that I could. I let that battery at the center, they've put it on recharge, today was the second day and it is still on recharge. And in the meanwhile, I've borrowed a battery from a friend. Just that was way more discharged than mine. It showed 11.9 V and it couldn't crank the engine. I've started the engine up with a good battery and with the engine running, I've placed the flat battery on. It is huge (for diesel), 82 Ah and 820 A. With that battery on, the alternator proved me that it works fine now. I took the car on a drive outside the city, not a long drive, just about 20 km, and after that, that battery can and is cranking the engine. Yesterday, after 4 hours since turning off the engine (and not very long drives before), the battery showed 12.20 V. And I'm still using it untill mine will be charged.
While on this subject, I want to point out the very easy way that I've got the alternator out of the car. So far, on all internet, I did not find this proper way described, for Punto with AC. What is needed is acces from under the car. I could use a ramp for that and it was very ok. First, disconnect the battery. Then, the upper bolt that holds the alternator, I got to it from above, slipping my hand near the AC compressor, the part towards the windshield. The nuts and the other two bolts then, got them out from under de car. Next, I got out the bolt from the anti-balance engine mount and using a pry bar, I've pushed the engine towards front of the car and I could get the engine out, downward, next to the CV shaft. And through there I've slipped the alternator back on, after job on it was done.
I put relevant photos for all of the above, hope you can work out which is what.
PS: I did not soldered the wires onto diodes bridge, just held tight together into bridge's support place. Also, I bought a new collector, have it on the car and I will put it in the alternator when need be.
The reason one slip ring is thinner than the other is fine dust caught between ring and brush and acting like a very fine grinder. There was the whistling coming from, too.