You're correct, sawing off this little material isn't possible by hand under normal circumstances. You could possibly do it working very carefully using a saw with a very thin plate (e.g. a Japanese saw) but it would still be challenging to saw off a strip this thin in a single unbroken piece. And once you get a breakage it's very difficult to restart a cut.
The correct tool for the job is a plane. Your smoothing plane isn't the ideal one for this, but it can comfortably do the job especially on a board as small and thin as this one.
TBH this board is so small that you could probably adjust for a heavier cut than normal and just set to work! But here's some general advice on setting up a plane for removing more material than normal that may be useful in future situations:
- Make sure your iron is extremely sharp; might as well freshly hone it before you begin. There's no such thing as too sharp with woodworking tools :-)
- Back the cap iron away from the edge 3mm or more IF your plane allows; not all do1.
- Adjust for as heavy a cut as you can comfortably take in whatever species this is2.
- If by chance you have the frog set forward of the default position (in line with the rear of the mouth) adjust it until it is.
- When you are close to your gauge lines set your plane back up the way you had it before and proceed more slowly until you're done.
Since a smoother isn't the ideal plane for this which one is? If we imagine a woodworker with a really comprehensive collection of hand planes a scrub or roughing plane is probably not what they'd reach for to remove ~2mm from the edge of a board (even one much thicker than this). Instead I think it's much more likely they'd use a jack or fore plane3 set up the traditional way — with cambered irons fitted. Note that this refers to the profile of the edge, not the bevel shape (bevels can also be cambered, as taught by Paul Sellers for example).
You might find some information in these previous Answers useful:
Different ways to set up a Number 4 bench plane
hand plane controls (bevel down)
1 The position of the slot in the cap iron and the adjustment mechanism in the plane sometimes heavily restrict how far back the cap iron can be set while still allowing the cutting edge to not to project too far.
2 You can take really heavy, 'woody', shavings working along the grain even in some hard hardwoods, but not in particularly hard ones such as some tropical species.
3 If not using wooden planes, this would be a 5 and a 6 in the Stanley numbering convention.