Ultra thin stiff wire
A needle may not be as stiff as you think. One reason a needle seems so stiff is that it's short. A needle of the same diameter and material that was a foot long would not seem stiff at all by comparison. This is just the mechanics of materials at play.

The design problem is a trade-off between wire diameter and material strength. The stiffness of the wire is increased with the wire diameter and also by the strength of the material; a wire of stronger material will deflect less than one of a lesser strength, given the same diameter.

The strength of steel is strongly affected by the alloy and its state of hardening. At the low end would be baling wire, which is mild steel (low-carbon) which cannot be hardened. At the high-end would be tool steel (chrome-vanadium alloy perhaps) that was hardened to maybe 60 Rc (really darn hard). These would exhibit really different deflection characteristics, from bending (permanently) with mild hand pressure to really making you work for it and probably snapping before taking too much of a bend.

A generally good compromise on these is available in music wire, as Matt mentions. Music wire is high-ish carbon steel hardened to a spring temper. This makes it good for springs, which is what a piano string is. It can be made harder, but this comes at the cost of increasing its brittleness. (If you snap a good needle with pliers you'll find that it snaps suddenly, aka. a brittle failure.)

Additional strength can be obtained using a tool steel, which I expect what the best needles are made from. This is most easily obtained as so-called "drill rod". It is supplied unhardened (but still quite strong) and can be hardened to your specifications using a propane torch and either water or (vegetable) oil.

None of these options will be absolutely rigid over any distance. You'll need to decide where to trade off a graceful curve with wire thickness. If you can prevent the wire from rotating, you may be able to "pre-stress" the wire (bend it upwards) such that it appears quite straight when loaded.

I would start with the music wire. It's much cheaper and easier to find. If it's important enough and music wire won't get it done you can look into tool steels.