Is it permissible to have sexual relation with someone not married to you, such as someone your right hand possesses?

Question

Is it permitted in Islam to have sexual relation with someone who is consent and willing yet not being in marriage bond?

Answer 1

ملكت أيمانكم (Malakat aymanukum) means those who you own, i.e. slaves. Refer to 4:25, 24:32, 30:28, 16:17, 16:75 etc.

It is permissible for a Muslim man to have sexual relations with a slave woman that he personally owns, without requiring a marriage contract (Nikah) since the exclusive ownership and authority substitutes for that.

And they who guard their private parts. Except from their wives or those their right hands possess, for indeed, they will not be blamed.

^{— Quran 23:5-6}

If a child is born to the slave from her owner, then she becomes an Ummal Walad (mother of the son) and can never be sold and is automatically freed on the owner's death.

Similar to marriage, no two men can simultaneously keep relations with a slave woman. It is not permissible to have sexual relations with a slave that someone else owns, unless you formally marry them; and if someone marries them then it is not permissible for the owner to have relations with them.

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Regarding your question:

Is it permitted in Islam to have sexual relation with someone 
who is consent and willing yet not being in marriage bond

The answer is No. Such a person will neither be your spouse nor your legal slave, but what you desire is something beyond that and hence verse 7 of Surah Al-Muminun applies:

But whoever seeks beyond that, then those are the transgressors

^{— Quran 23:7}

It would be Zina (fornication\adultery) and will not protect the rights of the partners nor the rights and lineage of any potential children.