[Jarek Duda (OP)]

Time crystals first described by Frank Wilczek in 2012 have got a lot of attention, recent popular review: https://physicsworld.com/a/time-crystals-enter-the-real-world-of-condensed-matter/

If I properly understand, they would like a lowest energy state spontaneously self-organizing into a periodic process - and propose sophisticated e.g. solid state experiments, or ping-pong of Bose-Einstein condensate, which don't really seem to satisfy this defining requirement (?)

But Louis De Broglie has already postulated in 1924 that with electron's mass there comes some ≈10^21 Hz intrinsic oscillation: E = mc^2 = hf = hbar ω, obtained if using E=mc^2 rest mass energy in stationary solution of Schrödinger's equation: ψ=ψ0 exp(iEt / hbar). 
Similar oscillations come out of solution of Dirac equation - called Zitterbewegung ("trembling motion"). Here is one of its experimental confirmation papers - by observing increased absorption when ticks of such clock agree with spatial lattice of silicon crystal target: https://link.springer.com/article/10.1007/s10701-008-9225-1

Electron can be created together with positron from just 2 x 511keV energy of EM field - after which it (the field?) should self-organize into these ≈10^21 Hz intrinsic oscillations.

So can we call electron an example of time crystal?
What other examples of lowest energy state self-organizing into periodic process are there?

ps. Beside self-organization of the lowest energy state into periodic motion (I don't see they got? in contrast to electron), they alternatively want "period doubling": that system oscillating with T period, self-organizes into 2T period process - breaking discrete time symmetry (invariance to shift by T).
So these popular Couders' walkers recreating many quantum phenomena in classical systems (slides with links) also have period-doubling (can they be classified as time crystals?) - here is such plot from this paper, horizontal axis is time, lower periodic process is for liquid surface - externally enforced by some shaker, upper periodic process shows droplet trajectory - self organizing into twice larger period than enforced:


But generally it seems very valuable to find analogies between spatial and temporal phenomena like crystals here.
Great tool for that is Ising model: Boltzmann ensemble among spatial sequences, what mathematically is very similar to Feynman path (temporal) ensemble of QM - using this mathematical similarity, for Ising model we get Born rule, Bell violation, or analogues of quantum computers.
What other phenomena can be translated between spatial and temporal dimensions?