Could some astronomical objects have superconducting properties? The colder it is, the more efficient the superconductivity process works. And as we know, if there is no star nearby,  space gets pretty cold.
I do appreciate that many condensed, burnt out, stars may take a long time to cool off, but are there any other types of known astronomical objects that may feature superconductivity to create and/or maintain a very strong magnetic field?
 A: Doubtful you'll find anything within the Solar System, but there are neutron stars, which are thought to have regions which are both superconducting and superfluid (that link is one of the original references from almost 50 years ago - there is a ton of literature on the topic since, you could start with some of these).
A: This is not a complete answer to the question, rather a explanation of Kyle Oman's answer.
When we (or at least me) think of superconductivity, we have in mind the pairing of electrons to form Cooper pairs. But this pairing is quite weak, and a moderate magentic field can destroy superconductivity.
But electrons are not the only particles around! At the extreme densities found in neutron stars, both neutrons and protons can form pairs. For them, critical temperatures can be as high as $5\cdot10^8$K for neutrons (larger for protons), and critical magnetic fields of $10^{15}$G. In neutrons, this pairing leads to superfluidity, and in protons to both superfluidity and superconductivity.
The protons of the outer core are thought to be in a Type II superconductor, that is, the magnetic field is confined to vortices where the field strength can be that of a magnetar.
And how has all of this been discovered? Neutron stars where discovered which cooled down unusually fast. This cooling wasn't compatible with their X-ray emission, so they are thought to emit neutrinos as well. The pairing of two neutrons lowers their energy, and this energy difference is liberated as neutrinos. Proton superconductivity is required to suppress other cooling mechanisms.
Sources


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*D. Page et al.: Rapid Cooling of the Neutron Star in Cassiopeia A Triggered by Neutron Superfluidity in Dense Matter arXiv:1011.6142

*P. S. Shternin et al.: Cooling neutron star in the Cassiopeia A supernova remnant: Evidence for superfluidity in the core arXiv:1012.0045

*C. O. Heinke: Superfluids and superconductors in the core of a neutron star: the highest-temperature superconductor University of Alberta

*B. Haskell et al.:  Investigating superconductivity in neutron star interiors with glitch models arXiv:1209.6260
