What is the punishment for a mother who kills her own child (aged 1-5 years)?

According to the Maaliki School of law, the punishment is death. However, most scholars hold that parents are not to be killed for killing their children.

The majority opinion is based on authentic ahadeeth in which the Prophet (ﷺ) said so.[1]

Reference:

[1] [Jaami’ at-Tirmithi, hadeeth number 1401.]