There is no literal mother - it is a symbolic reference to show God tender compassion and grace. The context is Israel's sinfulness and how they being treated because of their sin. God is saying that despite appearances, they will not be abandoned.
Barnes is most helpful here:
Where is the bill of your mother's divorcement? - God here speaks of
himself as the husband of his people, as having married the church to
himself, denoting the tender affection which he had for his people.
This figure is frequently used in the Bible. Thus in Isaiah 62:5 : 'As
the bridegroom rejoiceth over the bride, so shall thy God rejoice over
thee;' 'For thy Maker is thy husband' Isaiah 54:5; 'Turn, O
backsliding children, saith the Lord, for I am married unto you'
Jeremiah 3:14. Thus in Revelation 21:9, the church is called 'the
bride, the Lamb's wife.' Compare Ezekiel 16:See Lowth on Hebrew
poetry, Lec. xxxi. The phrase, 'bill of divorcement.' refers to the
writing or instrument which a husband was by law obliged to give a
wife when he chose to put her away. This custom of divorce Moses found
probably in existence among the Jews, and also in surrounding nations,
and as it was difficult if not impossible at once to remove it, he
permitted it on account of the hardness of the hearts of the Jews
(Deuteronomy 24:1; compare Matthew 19:8). ...
In the passage before us, God says that he had not rejected his
people. He had not been governed by the caprice, sudden passion, or
cruelty which husbands often evinced. There was a just cause why he
had treated them as he had, and he did not regard them as the children
of a divorced wife. The phrase, 'your mother,' Here is used to denote
the ancestry from whom they were descended. They were not regarded as
the children of a disgraced mother.