Can we know if an exoplanet has a magnetic field? The recent discovery of the exoplanet Proxima b gave rise to several questions concerning its habitability and its potential to host life. A major concern is the huge amount of radiation coming from flares emanating from its star. The planet is closer to its star that the Earth to the Sun and the flares seem to be more frequent there than here. Therefore, for this planet to be habitable -or actually inhabited by some life forms- a strong magnetic field is neccesary.
Thus my question: is it possible to find out whether an exoplanet has a magnetic field and, in such a case, measure its strength somehow?
 A: There are certainly many ways to go about detecting a magnetic field on an exoplanet. Many involve studying interactions with the home star. You may have heard of the case of HD 209458 b. Here, Kislyakova et al. (2014) looked at Lyman-$\alpha$ absorption around the planet, which was indicative of neutral atoms moving at high velocities. The only model that explained the behavior was a magnetic field with a magnetic moment about that of Jupiter's, interacting with the stellar wind.
There are other signs of a planet's magnetosphere. Electrons traveling along field lines in the magnetosphere can lead to radio emissions from electron cyclotron radiation, which has been observed in the gas giants in the solar system. Detecting these emissions from exoplanets is a lot trickier, as the signals will be fainter. However, at distances of less than 20 parsecs, this method might be feasible in the near future.
Yet another possible technique would be to try to study bow shocks (see Vidotto et al. (2010)) caused by the interaction of the planet's magnetosphere with the stellar wind. This - as well as the other methods - can work extremely well when the planet is close to its home star, as is the case with Hot Jupiters. You can also study atmosphere loss on these planets (which would be influenced by the presence of a magnetic field) - which is similar to what was done with HD 209458.
There's also the curious case of HD 179949 b. In 2004, astronomers noticed a hot spot on the surface of HD 179949, the host star. The spot has the same period as the planets, and appears to be the result of an interaction between the magnetic fields of the exoplanet and the star. I am not aware of newer observations of the spot, however.
I'd strongly recommend taking a look at Chapter four of this book for a brief summary of the different detection techniques, as well as a discussion of theoretical scaling laws.
It's also worth noting that Barnes et al. (2016) created models that found that it's possible for Proxima Centauri b to have a magnetic field, although there's not yet data to confirm or refute the possibility. Barnes himself explains that, as well as other habitability issues, on the Pale Red Dot website.

I'm aware of two recent papers (Vedantham et al. 2020, Pope et al. 2020) that claim an indirect detection of an exoplanet orbiting the star GJ 1151 by 1) detecting radio emission at ~150 MHz and 2) ruling out companions of $M\sin i>5.6M_{\odot}$, theorizing that the signal is from auroral emissions to due an exoplanet's magnetosphere. I can say more about that once I've taken a good look at the papers - and perhaps after anyone is able to make independent observations of the system.
