Is it permissible, Islamically, for the sister to demand her share of the inheritance after the father's death, given that she received her share from him a year before his death, without a contract or proof?

There is no disagreement concerning a daughter's right to inheritance; she receives half the share of her brother after the fulfillment of obligatory shares. What a father does during his lifetime by dividing the estate does not establish a right for the daughter in the estate until after his death, because the living do not inherit. If the father divides his property among his children as a lawful gift (hiba shar'iyya) and it meets the conditions, then this is permissible. However, if the division is without lawful transfer of ownership, it is void, and everything the father owns reverts to an estate to be divided, with the male receiving the share of two females.

As for what the daughter took from her father, assuming it was her right in the inheritance: if it was a gift and she took possession of it during his lifetime for a just cause, such as need, then she is not required to return it. But if it was taken as her right in the inheritance, then it is void and must be returned to the estate, and she is not permitted to take more than her right except with the consent of the other adult and mature heirs.

Matters of inheritance should be referred to the Sharia courts for investigation, as there may be wills, debts, or rights that take precedence over the heirs' rights. The estate should not be divided without consulting the Sharia courts.