Atoms are the only limit, as Arkcon pointed out. And if a thin tip is more interesting than a thin cylinder, you have a single atom at the sensing point of a tunnel effect microscope or an atomic force microscope.
You have some intermediate diameters between drawn steel wire and nanowires. For instance the usual carbon fibres used in composites have commonly 4µm to 7µm diameter. Gold wires used (less and less) to bond silicon chips are but thicker. Beams of silicon can be made narrower by semiconductor processes. Asbestos fibres split down to some 20nm diameter.
One limit is stiffness, yes. No material is magic here: steel is already excellent, a few ceramics are just 2× stiffer, nanotubes 5×https://en.wikipedia.org/wiki/Young's_modulus
So the diameter makes all, and the stiffness of a cylinder (or of any beam) varies as the thickness high 4. If comparing at identical length, the limit is close. I didn't use steel wires, but steel sheets of 0.1mm are already quite flexible, so you couldn't push with a good human force on a 0.1m long sample.
On the other hand, if the needle's length scales down like the diameter, then there is no such limit, because a beam's stiffness varies as the length high -3 too. You may also refer to beam buckling by compressionhttps://en.wikipedia.org/wiki/Euler%27s_critical_load
which tells that, if the length and diameter vary by the same amount, then the buckling force varies as much as the cross-section area does, just like the limit by the material strength does.