Are all elementary particles indivisible?
If by elementary particles, you are including protons and neutrons, then the answer is no. The reason for this is because protons and neutrons are composed of even smaller particles called quarks.
There are six distinct 'flavours' of quark. These are: up, down, top, bottom, strange and charm. Protons and neutrons are composed of up quarks and down quarks, while the other four 'flavours' of quark are unstable, and so it is less likely that they will combine to form composite particles, or 'hadrons'.
It is unknown whether there are elementary particles that are smaller than quarks, as it was first thought, before the discovery of protons and neutrons, that the smallest elementary particles were electrons.
The electron was discovered in 1897, the proton was discovered in 1918, and the neutron was discovered in 1937.