I found that neither of the Baofeng branded 3800mAH BL-5L batteries, bought from two different vendors, would securely latch in either of my UV-5R transcievers. Maddening enough, one of the batteries came from China packaged with a UV-5R and desk charger as its only power source.
If you're having issues with your 3800mAH battery clicking into place, first, verify you have the correct battery. The definitive site for the UV-5R (among other Chinese radios), is Miklor. On that site is a UV-5R FAQ, and a section on batteries. You will find a photo of the different case bottoms, there are differences, and if you bought the wrong battery you can either return it and buy a universal one, or try the "hold my beer and watch this" "grind it 'till it fits" method available on a YouTube video where a guy is attacking his battery with a Dremel tool. You'll have to search for that yourself, I won't provide a link and be party to you putting your eye out.
First, I'll define my nomenclature. I'll call the Battery Release button (marked "PUSH") the Latch.
I'll call the extrusion that pushes up on the Latch when you slide the battery onto the radio and should allow the Latch to drop down and securely hold the battery in the Hook. I'll call the smooth extrusion above and around the textured shoulder of the battery the Rib.
When I checked my stock BL-5 1800mAH battery with a #51 drill bit, the bit would fit under the Hook and above the Rib. If you don't have numbered bits, try it with a slightly smaller 1/6" (.0625") fractional. The same drill bit would not fit the same way on either BL-5L.
The solution is to remove enough material from the bottom of the Hook so that the Latch can engage. I did this with a flat Swiss needle file from Enco. Harbor Freight has a similar set, but they appear longer, so I can't vouch for their thickness. I measured my file at around .056" thick, and it would not fit between the Hook and Rib of the BL-5L either.
So, if you do the math you can see the Hook is a bit over .011" out of tolerance. I carefully worked the file against one side of the Hook with the battery pointed upwards in my hand, then pointed the battery down and worked the other side of the Hook until I could get the file straight, then applied pressure against the Hook to finish up, using the drill bit as my reference gauge as I went along so I didn't get over zealous. The trick is to stay away from the Rib since it's not causing any of the problem.
I tested in the radio and with mild upward pressure I could get the battery to click into place. I took just a wee bit more material off until no pressure was needed.