Say from a standard board 1 in x3 in x6 feet cut strips of 1/8 in thickness?
Adjust your thinking.
You've noted that you're trying to make some trim pieces. I can't, for the life of me, envision what you need 1x1/8" trim for, but I'll assume you need it and that's the perfect size for what you're after. (Oh! You're making doll-houses or other scale models! Of course.)
I'd suggest that you rip your wood in 1/4" wide strips, then plane to the final thickness.
Even the highest quality hand-saws will leave a rough cut that won't be acceptable for finish trim work. By cutting to 1/4" thickness, your cutting will be easier to handle than it would be at 1/8" (though it will still require a level of precision that you'll likely only attain on the later strips after you've had some practice).
By planning ahead to plane it to the final thickness, the precision of your cuts won't matter nearly as much, and the planed surface will be much nicer than the rough-cut surface. Depending on the installation details, you can get away with only planing the visible side. You may need to give the hidden side a pass or two with the hand plane to get it smooth enough to sit nicely, then do all the finishing work on the visible side.
A properly sharpened and adjusted hand plane will give you paper thin shavings, somewhere on the order of 1/64", so you'll be able to sneak right up to your final thickness. I'd suggest popping over to Woodworking.se for loads of info on sharpening and tuning the plane to get it to make these paper thin shavings.
Depending on how many pieces you need to make, I'd recommend a jig into which you can place the freshly cut pieces, with a stopper at one end to prevent it from moving, and sides to act as a depth-stop so you don't accidentally over plane and end up with something too thin.
As I understand it, ripping 1" x 3" x 6' to 1/8" thickness would result in strips that are 1" x 1/8" x 6'. I think that will be very hard to do with hand tools.
My recommendation is start with 1/8" thick wood. Not a lot of choices, but at least some available in 4' x 8' sheets. Starting with that, you will be cutting something like 2' x 6' x 1/8". The cuts will now be 1" wide instead of 1/8" - much easier to handle. Each cut will be through 1/8" of wood instead of 1" - so it will be 8x as fast & easy - actually a lot easier - almost to the point of score & cut.
Guitar makers do this all the time, but they don't use a saw, generally. Create a jig that holds a chisel as a scraper, and then you pull the strips through it, narrowing the space slightly each time.
The idea is to saw oversized and then plane or scrape to thickness. The takeaway here is that 1/8inch strips are best thought of as "binding". And binding has traditionally been made with a scraper.
As for how to get the starter strip, that depends on the kind of wood. Again, guitar makers would probably be doing this with softwood, and the initial cut would just be careful successive cuts with a sharp knife along a scribed line.