Long Jigsaw Blades for wood

I don't think a jigsaw would be suitable, it will lack power and a long blade will wander and probably break.
You really need a circular saw, chainsaw or even a decent hand saw.

 

As above a decent handsaw has become my go to for cutting 'sleepers' (they're not but that's what they're sold as) and I've got pretty much every type of saw available - chop saw, recip saw, large circular saw chainsaw, bandsaw, table saw, track saw oh and jigsaw of course. Of that selection I'd use the bandsaw for long rip cuts and the one I most definitely wouldn't use is the jigsaw I'm afraid.
If the cuts aren't going to be visible I might consider using a chainsaw but they're such a faff that I'd have to be doing loads to warrant it.

 

At 135mm a circular saw won't cut all the way through, but I'd start the cut with that and then finish with a handsaw keeping the blade snug to the side you're keeping. If you're keeping BOTH sides, then handsaw all the way so the cut is uniform.

 

If they really are real railway sleepers, as opposed to pressure treated softwood timber sold as sleepers, then you won't have any joy out of doing it with a handsaw.
I have a whole retainer wall, garden bed arrangement that I made from real sleepers (the ones that still have the odd bolt holes in them from where the rails were attached). I cut many of them ... sometimes for lap joints, sometimes diagonally, and others with just plain right angle cuts to length. I used my hand-held circular saw to cut the top to the full depth of the saw blade. I then marked it carefully and did the same on the underside of the sleeper. There was then only about 40mm or 50mm of timber left between the two cuts and I cut that out with my handsaw. It was really hard work ... so much so that I gave myself tennis elbow from all that sawing!

 

As Roger says "proper" sleepers are a whole different can of worms and if reclaimed will often have stones stuck into them worth having a scrape along the line of the cut to get rid of the worst of them before you cut with whatever weapon you choose.

I would use chain saw every time: you can make accurate cuts with a bit of practice.