Great pistols.
I have two. And mine are both the "bad" old design that Browning actually recommended disassembling and cleaning.
There's also a pristine Browning Challenger in the family (precursor to the Buckmark/Challenger II), that is an excellent pistol and highly praised by everyone that uses it.
I also bought a Buckmark rifle the day they hit the streets (1999?). Fantastic. Awesome. The funnest rifle I have ever owned. But I sold it...
One of these days, I'll find a Ruger Mk I that makes me happy (also good pistols). But, until then, it's this pair of Buckmarks and the once-a-decade chance to shoot the Challenger.
One of the Buckmarks (a mid-'80s "Standard") has been discussed several times on TFL. Current round count is 130,620 rounds. It has seen so much use that the slide and barrel are slightly mushroomed from the slide slamming into the barrel 140,000+ times. It's on its fourth or fifth recoil guide rod assembly (includes rod, spring, retainer, and guild block), and has gone through several dozen buffers, but that's it. No parts failures, just routine maintenance. Oh... that's not true. I wore out a magazine spring and follower that came with the Buckmark rifle.
The originals are still going strong with no parts replacements - just maintenance.
The other Buckmark, an early Target model (early version of what Nathan posted), had a much harder life and round count is difficult to nail down. I estimate 120k, but it's anybody's guess. It broke a firing pin after considerable abuse, and still ran for 10k rounds or so with the broken pin; but hasn't had any other parts failures.
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"Those screws! Oh, those screws!"
Lefty-loosey, righty-tighty. Read the instructions, and everything is fine. And you don't have to remove both (or all three, or four) top strap/rail screws. Remove the rear screw and loosen the under-barrel screw. That's it. Pull the slide back slightly and pivot the barrel assembly forward and up to clear the frame. Only ONE screw has to actually come out of the frame, and it won't require sighting-in the pistol again. (And if you have the older slotted head screws, upgrading to the newer hex screws is well worth the $5.)
Some time in the '90s, Browning changed a few parts, including going to the self-contained firing pin assembly that drops into the slide (rather than firing pin being pinned into the slide and retaining the spring itself). Along with those design changes came disassembly and cleaning changes. Before, Browning recommended disassembling and cleaning on a regular basis, and specifically stated that replacement of the 'rib washer' under the heads of the top strap screws was mandatory every time the screws were removed.
But, after the design changes, Browning no longer recommended any disassembly by the end-user. They went to the "spray it out and it'll be fine" suggestion.
The problem with screws getting stuck or backing out seems to arise from either not reading the instructions or having the later versions of the instructions that do not tell you that the washers are meant to be one-time-use. (Or the factory failing to install them, to begin with.) If you don't replace the washer(s) at least every second or third disassembly, the screws will A) get stuck, or B) loosen up with use. I keep about a dozen spare washers on hand and replace them when I can no longer feel the 'squish' as the screw is tightened (about every other disassembly).
The screws never work loose, nor do they seize up.