Are there experimental observations of the Abraham-Lorentz force? The Abraham-Lorentz force is the force a classical charged particle particle exerts on itself due to its own electromagnetic field. It has a rather simple formula that reads
$$
\vec{F}_\mathrm{AL} = \frac{2 q^2}{3 c^2} \vec{\dddot{x}} \,.
$$
My question is the following: Is there any realizable context where a classical charged body can be observed to experience the Abraham-Lorentz force?
It is unlikely that one could directly observe the immediate acceleration due to $\vec{F}_\mathrm{AL}$, but an indirect observation through long-term energy losses of the body might be observable.
 A: Ok, I would like to thank Andrew Steane and Vladimir Kalitvianski for their input. I have done some digging myself and I believe I have gathered enough material to compile an answer to the question from the following sources:

*

*The 2017/2018 notes of Kirk T. McDonald On the History of the Radiation Reaction,

*the 2016 paper by Di Piazza et al. Investigation of classical radiation reaction with aligned crystals, and

*the 2017 paper by Wistisen et al. Experimental Evidence of Quantum Radiation Reaction in Aligned Crystals.

According to McDonald:

We noted earlier that while the radiation reaction for oscillating currents has clear manifestation in the so-called radiation resistance of antennas, there is no experimental evidence
for the classical radiation reaction of an individual electric charge.

Now let me cite the abstract of Di Piazza et al.:

The  self-consistent  underlying  classical  equation  of  motion  including
radiation-reaction effects,  the Landau-Lifshitz equation,  has never been tested experimentally,  in spite of
the first theoretical treatments of radiation reaction having been developed more than a century ago.  Here
we show that classical radiation reaction effects, in particular those due to the near electromagnetic field, as
predicted by the Landau-Lifshitz equation, can be measured in principle using presently available facilities,
in the energy emission spectrum of 30-GeV electrons crossing a 0.55-mm thick diamond crystal in the axial
channeling regime

By the Landau-Lifschitz equation they mean the approximation where the term $\dddot{x}$ is replaced by the jerk felt by a hypothetical "test particle" accelerated in the very same external field but without any radiation reaction. (I.e., simply an approximation of the Abraham-Lorentz force.)
The proposal of Di Piazza et al. was then experimentally realized by Wistisen et al. but they found that the classical approximation of the radiation reaction is not a valid model and that:

The measured photon emission spectra show features which
can only be explained theoretically by including both 1) quantum effects related to the
recoil  undergone  by  the  positrons  in  the  emission  of  photons  and  the  stochasticity  of
photon emission, and 2) radiation-reaction effects stemming from the emission of multiple photons.

2021 Update: A follow-up paper by Nielsen et al. from 2020 under the title Radiation Reaction near the Classical Limit in Aligned Crystals showed that

Hitherto, the experimental problem in validating the LL equation has been to achieve sufficiently strong fields for radiation reaction to be important without quantum effects being prominent. Notwithstanding, here we provide a quantitative experimental test of the LL equation by measuring the emission spectrum for a wide range of settings for 50 GeV positrons crossing aligned silicon single crystals near the (110) planar channeling regime as well as 40 GeV and 80 GeV electrons traversing aligned diamond single crystals near the ⟨100⟩ axial channeling regime. The experimental spectra are in remarkable agreement with predictions based on the LL equation of motion with small quantum corrections for recoil and, in case of electrons, spin and reduced radiation emission, as well as with a more elaborate quantum mechanical model.

That is, it can be safely stated that at least the behaviour of beams of particles is consistent with the Abraham-Lorentz (Landau-Lifschitz) radiation-reaction force whenever the classical limit is applicable.

Thus, as far as concerns the original question, we can state that there exist:

*

*Experimental realizations where ensembles of particles feel radiation reaction that can be modeled by the Abraham-Lorentz (or Landau-Lifschitz) formula. This includes radiation losses by beams of particles in particle accelerators or crystals, as well as the "radiation resistance" of currents in antennas.

*Experimental realizations of quantum radiation-reaction that cannot be reasonably modeled by the Abraham-Lorentz (or Landau-Lifschitz) formula.

However, there currently seems to be no realizable experimental setup where effects of the Abraham-Lorentz force on a single classical body can be measured or observed.
A: The Abraham-Lorentz force is wrong since it leads to non physical solutions. It is not used in calculations. In practical calculations it is replaced with another force, namely with $\tau_0\dot{\vec{F}}_{\text{ext}}(x,t)$, see Landau-Lifshitz textbook and [1]. This force does not conserve the energy-momentum, strictly speaking, but it is much better option than the pure Abraham-Lorentz force since it behaves more physically: it indeed takes (approximately) into account radiative losses, which are obviously present in reality (in betatrons an other accelerators, for example, as well as in collisions).
[1] Fritz Rohrlich, The dynamics of a charged particle, (2008) http://arxiv.org/abs/0804.4614
