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Though it is highly unlikely, has any carbon compound been found to make an ionic bond and to exhibit ionic properties?

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Why unlikely? Ionic compounds of carbon have been known for ages. There are ionic carbides ($\ce{Al4C3}$, $\ce{CaC2}$, etc.), graphite intercalation compounds like $\ce{KC8}$, ionic derivatives of fullerenes and more.

Come to think of it, common $\ce{CaCO3}$ is certainly ionic and at the same time a compound of carbon, but this is most likely not what you want.

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There is also the classical case of the aromatic carbocations and carbanions; examples include tropylium bromide,

tropylium bromide

and sodium cyclopentadienide, which is used for preparing "sandwich compounds", among other uses.

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Perhaps what you meant to ask was

Are there any known organic compounds with ionic bonds?

Organic salts would be examples of organic compounds with ionic bonds. Here is a list of organic salts.

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I think, what actually OP meant to ask is:

Are there any known organic compounds with carbon involving in an ionic bond?

The reason for my suggestion is the phrase "to make an ionic bond and to exhibit ionic properties" in OP's question, even though it did not describe the involvement of carbon. Otherwise, any ionic carbonate (e.g., $\ce{Na2CO3}$) or ionic cyanide (e.g., $\ce{NaCN}$) would qualify, none of which are not organic compounds.

Based on my suggestion tropylium bromide and sodium cyclopentadienide certainly qualify.

I think the best example is a compound containing triphenylmethyl cation (triphenylcarbenium or tritylium) and a non-nucleophilic anion (which doesn't have a strong affinity for bonding to the electrophilic carbon). Such compounds are stable at room temperature and also commercially available $($e.g., triphenylmethyl tetrafluoroborate, $\ce{(C6H5)3C+ BF4-})$:

triphenylmethyl tetrafluoroborate

Compounds of tritylium with other anions such as hexafluorophosphate $(\ce{PF6-})$, hexachloroantimonate $(\ce{SbCl6-})$, and perchlorate $(\ce{ClO4-})$ are also known. Further, perchlorotriphenylcarbenium Hexachloroantimonate $\left(\ce{C6Cl5)3C+SbCl6-}\right)$ has been prepared and obtained its crystal structure showing the positive charge on the carbon (Ref.1):

The title compound, $\ce{C19Cl15+.SbCl6-}$, consists of perchlorotriphenylcarbenium cations and hexachloroantimonate anions. The central $\ce{Sb}$ and $\ce{C}$ atoms occupy special position of symmetry 32 along the c axis. The cation shows a symmetrical propeller conformation. Four $\ce{Cl}$ atoms of the $\ce{SbCl6-}$ anion are disordered.

References:

  1. E. Molins, M. Mas, W. Maniukiewicz, M. Ballester, J. Castañer, "Perchlorotriphenylcarbenium Hexachloroantimonate(V)," Acta Cryst. C 1996, C52, 2412-2414 (https://doi.org/10.1107/S0108270196007287).
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