The VIC-II in the C64 outputs 320x200 pixels inside the borders. What is the full visible resolution including the borders?
There are no pixel outside the pixelated area. The border just fills the lines. And whats visible is - like always, defined by the adjustment of the monitor/TV.
To clarify, I am interested in the full screen size in units of normal pixels, i.e. my question has nothing to do with whether you could possibly change just a single pixel in the border area.
I belive your question is based on the idea to exactly reproduce a C64 output on a digital screen. Just TV never was digital, and even less an exact thing. There are no pixels and no pixel units. TV is defined by timing and asumption. And based around the fact that not two receivers will show the same portion of a transited picture. Therefore does every TV signal carry an overscan area. That's a the part of the picture that might be shown but doesn't have to be.
The whole thing starts with CRTs not beening rectangular in the first place and includes many other effect that will make less than transmited visible. Each show/movie is produced in a way that there is no relevant content in the direct border area - horizontal or vertical. For example the guideline for 'action safe' is that the action should hapen at least 5% from each side. At the same time, 'title safe' is set at 10%. So no title, text element or logo should cover an area less than 5% away from the sides.
And even in the age of digital TV this is relevant, as console manufacturers also set standards in that region for their products: Nothing game relevant (like controls) within 5-10% of the sides. That's why there's often a rather thick border around gauges.
On a C64 (and effectivly any other machine), a useful (pixelated) picture is produced within the line timing of a TV. Some keep the area outside of this at black shoulder level, others allow it to be filled with a single colour - sometimes even on a line by line base.
A whole TV line is defined (NTSC, PAL in braces) as 63,6(64) µs. Thereof 10.9 (12) µs are used for synchronisation purpose and 52,7 (52) µs are for a portential visible signal. How much of that is realy visible depends on the individual TV set (*1)
Ofc, one could now argue that of that by suptracting the time for the 320 pixels displayed we get the overscan area and by deviding this again by pixeltiming we get some pixel number. That gets tricky, so lets try:
A NTSC C64 runs at 1.022727 MHz, with 8 Pixels displayed per clock cycle. That's 0.122 µs per pixel. Thus a 52.7 µs line content (*2) does equal 431.18 pixel length.
For PAL the numbers are 0.98524 MHz and 0,127 µs per pixel, resulting in 415,38 pixel times.
So one could state that the maximum C64 line is made of 431 (415) pixels, thus the border would be 431-320=111 pixel (415-320=95).
Just, that it doesn't tell how much of these 'pixeltimes' are visible or not. On monitors (and dedicated TV) users usually blew up the picture until the corners of the square (useful) picture reached the round ones of the screen, wo get a maximum sized useful picture. Easy to see when comparing different pictures of C64.
Vertical its less complicated, as lines are already a quantitized resource. The content carrying portion of the first field (and homecomputers usually just produce first fields) ranges from line 20 (24) to 262 (310) resulting in 242 (284) displayable lines, thus it's a maximum of 42 (84) border lines (*3).
Again, noone blows a TV pic up to the full extend. Especially not when home computer output is displayed.
Bottom line: What's called a border on the C64 is everything outside the pixelated display area (but within the line/frame) and what part thereof is shown or overscan does not depend on any definition but is how an individual TV set is adjusted.
If you're abotu to do some emulation or whatsoever, just pick a nice number, like 10 pixels/lines, for your border area, ignoring if it's a NTSC or PAL machine. This will give a nice and consistant user experiance.
P.S.: There is a great page about real life VIC-II behaviour, even though his calculations are based on asumptions and a bit off, his notes about colour behaviour are quite interesting.
*1 - If you realy want to dig into the TV's signal structure, maybe check this very good (and compact) clue sheet.
*2 - Many people often use the total line length here for calculation, not just the content part. While this is neccessary when calculating C64 program timing, it's of no meaning about how much is dsplayable, as it includes all synchronizing parts.
*3 - As a result, with a 'standard adjusted' TV-screen pictures of European C64 output should show a wider top and bottom margin than US models. Except, many promo pictures of commodore, that clearly show a European models output (no photoshop back then), don't show such ... as if they cranked up the picture to get a better image. Eventually the ulimate proof how irelevant that calculations are.