Haven't seen any evidence of that. Pakistan claimed to hit the 96L6E “Cheese Board” early warning C band (500MHz to 1000Mhz) radar at Adampur Airbase, using CM-400AKG Air Launched Ballistic Missiles (ALBMs), with their passive radar seekers modified for the anti radiation mission. This radar can be mounted on a wheeled chassis, or on an elevated observation mast that gets towed around for better low altitude coverage.

Satellite imagery showed some movement at Adampur, but no signs of explosions at previous radar sites. India admitted to minor damage at the base, but said that the S-400 system was intact. So it looks like the S-400 survived unscathed, but failed to intercept all projectiles. All in all neither system performed great, but the S-400 did outperform the HQ-9. The ALBM threat to these systems, US, European, Russian or Chinese is very real, even ground launched ballistic missiles like ATACMS have destroyed components of the S-400 system before. To understand this, we must look at the missiles used.

The primary anti ballistic missile for the S-400 is the 9M96 and 9M96E2 missile, a Russian equivalent to the Patriot PAC-3 MSE. A kinetic hit to kill interceptor with a thrust vector control altitude control system (ACS). Think of these missiles as mini Talon missiles from the THAAD system, with a much lower maximum altitude. (20-35km vs 150km). These missiles, like Patriot PAC-3 MSE, are considerably smaller than the blast fragmentation missiles for aerodynamic targets, as seen in the below photographs. Here three 9M96 missiles take up the same space as one blast fragmentation missile.

Patriot adopts a similar solution for a combined loadout, with the smaller PAC 3 missiles taking up less space than the PAC 2:

The problem with these combined loadouts is that magazine capacity is compromised between engaging aerodynamic and ballistic targets. The Russian solution to this is the S-350, an air defence system that just uses the 9M96 missile. South Korea has it's own version of this. The US solution is a Patriot battery armed just with PAC 3. Think of these systems as mini THAAD, just for engaging shorter range ballistic missiles, they can engage aerodynamic targets, but it's not their primary mission. One S-350 launcher can carry 12 9M96 missiles:

Same for this Patriot launcher exclusively with PAC 3 missiles:

Theoretically an S-400 can also be kitted out just with 9M96 missiles:

But ultimately S-400 (initially named S-300 PMU-3) and Patriot were primarily built around engaging aerodynamic targets, their blast fragmentation missiles can engage ballistic targets, but the results are suboptimal. The kinetic hit to kill interceptors were added afterwards to improve ballistic missile interception, but there are still compromises in magazine capacity and in the sensor chain for engaging ballistic targets, especially modern ballistic missiles that can do pull up manoeuvres and so forth. The Soviet Union and Russia nowdays has S-300V, a system built around terminal interception of ballistic missiles and low altitude cruise missiles, using bespoke radars, massive interceptors with a 150kg blast fragmentation warhead to intercept ballistic missiles. But this system is even more expensive than S-400. Some S-400 batteries in Russia have been adapted to use the modernised missiles from S-300V4/S-500 to further improve anti ballistic missile capabilities. Also Russia seems to have production issues with the 9M96 missiles, just look at all the public photographs of the S-400, how many have 9M96 missiles pictured? How many S-350 batteries exist? Russia placed an order for a total of 12 S-350 batteries, for 144 launchers and and 1728 missiles, between 2018-2027. That's around 170 missiles a year, or at best 250 a year if deliveries started in 2021, and only half of this order has been delivered. The USA produces 500-650 PAC 3 missiles a year.

I don't think it's really over unless Chinese hardware is cheaper and more abundant as well. Some countries were using t-34s for decades after they were obsolete just for price reasons.

Article here claims that an obituary for an S-400 operator was published in Bihar province. Pakistan claimed to have hit the radar and command post for the system rather than a launcher, which is worth keeping in mind. I wouldn't imagine one instance of a successful hit on the system will hamper Russian military sales though, because:

• There are a lot of factors that go into whether a strike is successful or not, and the quality of the AD system itself is far from the only one
• No serious military should be going into a war believing that any of their systems are invulnerable
• It's not like anyone else in the world makes a significantly better system than the S-400, especially at the price India is able to pay

Overall though it seems like India bit off more than they could chew with this confrontation.

I have no idea if it was actually destroyed, but no air defense system is perfect. Even Ansarallah was able to defeat Israel's incredibly dense and sophisticated network with a single missile earlier this month. War between peer countries will always incur losses.

Seems like arms deals are indeed based on vibes and propaganda (see HIMARS and now J-10C), so maybe it will hurt Russia's industry anyway.

did the Pakistani air force destroy India’s S-400 batteries using Chinese-made missiles?

According to gloating from Indian Twitter, Pakistan hasn't given real evidence that this happened. Make that for what you will.

PAF most likely did not destroy any S-400 battery. Any claims that it did arent any more substantiated than Indian claims that Chinese HQ-9 or HQ-17 systems were destroyed. Actualy my understanding is that the S-400s performed quite well and its the one system India immediately approached Russia after the conflict for more orders