Vanessa's mom, Sofia Laine, has filed a lawsuit against Kobe's estate, claiming Kobe promised to take care of her financially for the rest of her life.

In new docs, Vanessa points to a legal fight Sofia was having against her ex-husband in 2004 and 2008 over spousal support. Sofia's ex-husband claimed he should not have to pay her because she was being supported by Kobe and Vanessa. Sofia fired back that her daughter and son-in-law had no obligation to support her, and whatever they gave her was simply out of the goodness of their hearts.

Sofia's ex also pointed to tabloid stories which claimed Vanessa bought a $1 million home for her so she wasn't in need of support. Sofia shot back ... the stories were "absolutely false" and "I would never permit Vanessa to do such a thing. I have not and do not (nor should I be required to) rely on Vanessa for my support."

And, Vanessa has other reasons why she wants the lawsuit dismissed. Among them ... Sofia argues Kobe made an oral promise to "take care of her financially." Vanessa's legal team says even if Kobe said that, it's too vague to be enforced.

And, there's this ... Sofia also says Vanessa and Kobe violated California labor laws by not giving her meal breaks, rest periods and giving her minimum wage for babysitting services. Vanessa fired back in legal docs, "[Sofia] was never an 'employee' ... [she] is a grandmother who, at times, helped her daughter and son-in-law by spending time with her grandchildren."

And, Vanessa's legal team argues, in order for Sofia to go after Kobe's estate, she had to file a creditor's claim within a year of his death, and she did not do that.