This one, for the same model as you, 9560

https://smile.amazon.com/gp/product/B07DG1TZ4P/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

Mine had the same issue: trackpad became increasingly hard to depress in the front corners as mouse buttons -- and then started to lift up -- first visible issue. I searched online and found threads like this. Opened the case and saw the swollen battery.

I searched re: batteries, and no WAY would get a "refurbished" OEM battery from Dell. I wanted something new, with new LiIon cells, etc. In the end, I purchased one on Amazon. It has mixed reviews, but it seems that some of the negatives were from totally clueless customers. The people who sounded credible gave it good reviews -- and I had to option to return it if there were problems, so I went ahead with the order. Overall -- a very positive experience. I also posted a review of the one I bought on Amazon. Here's the review I wrote, along with info regarding the manufacturer of the battery and cells inside -- every bit as credible as the "OEM" version (which Dell subbed out anyway):

Product URL: https://www.amazon.com/SWEALEER-Compatible-6GTPY-Applicable-Replacement/dp/B07DG1TZ4P/

Review URL: https://www.amazon.com/review/R30NMDTVN87GLI/

Body of Review (in case you don't want to go there...):

Product arrived in 2 days as promised. Very nicely packaged. The box was clean and the battery was inside an anti-static enveloped sandwiched between 2 layers of spongy foam. There was also a separate zip lock bag enclosed with the torx and phillips drivers needed to remove the laptop cover and battery. The whole replacement took just a few minutes.

The only concern I have is that the original battery was spec'd at 97 wHr, and when I general a battery report in Windows 10 using the command powercfg /batteryreport /output "C:\battery_report.html" to place the report at the root of my C: drive, it looks like the max capacity of this battery reads as 91.2 Wh (91,200 mWh).

Anyway - the laptop booted up perfectly after the replacement -- no BIOS issues or any other problems. The battery arrived 60% charged, and in less than 2 hours with the laptop connected via USB-C / Thunderbolt 3 cable to my Dell Thunderbolt 3 dock, the battery is already at 90%.

I also checked, and it appears that this replacement battery is made by LGC (LG Chem). They're the successor company to Lucky / Goldstar, a $20+ billion conglomerate based in South Korea. LGC is also one of the primary battery suppliers to Lenovo for their laptops. The original battery that came with my Dell XPS 15 9560 was made by Simplo -- a Taiwanese company (I can see that on the printed section of the old battery). Simplo is known for providing batteries for Macbook's. I have no concerns about batteries from either company. I'd just like an explanation regarding my new battery showing up as having only a 91,200 mWh capacity, when the original was spec'd at 97 Wh, and this new one is listed here at 97 wHr.