In Naruto Shippuden episode 147, a technique used by the ANBU forces of Kirigakure called Kirisame no Jutsu made rain fall down that absorbed chakra, and dissolved Naruto's Rasengan, Utakata's Soap Bubble Ninjutsu and Sai's Ninpo: Choju Giga. Considering Amaterasu can't be extinguished with water, but this water absorbs chakra and other people's techniques, can the Kirisame no Jutsu technique extinguish the flames of the Amaterasu?
As @Rumpelstiltskin mentioned in the comments,
any technique capable of absorbing chakra can ideally extinguish the Amaterasu cause the jutsu itself is a form of chakra.
Amaterasu can practically only be stopped, if one, sealed, as Jiraiya did. Or, if absorbed. It will also extinguish if it passes its time limit of seven days and seven nights.
Amaterasu cannot be extinguished but sealed.
Taken from Quora page: What would Jiraiya do if Amaterasu hit him?
And, Amaterasu can also be defeated by absorbing jutsu, as Madara has shown when Sasuke cast Amaterasu on him. Although most of the flames were concentrated to his armor, he managed to remove it by taking his armor off and absorbing the rest of the Amaterasu with the Rinnegan's Preta Path.
The wiki also explains that:
The Preta Path (餓鬼道, Gakidō) grants the user the ability to absorb chakra in any form. ... This ability is primarily defensive in nature as it is capable of absorbing chakra from an individual through physical contact. (emphasis my own)
Any jutsu that can absorb jutsu ideally can absorb jutsu / dissolve any justu should be able to absorb Amaterasu, whether the technique used to dissolve the jutsu is made of water would not be much of relevance, as it can still absorb jutsu. Thus, it is likely that the Secret Technique: Mist Rain can absorb/dissolve Amaterasu.
But as @Makoto mentioned in his own answer, this jutsu has never been applied to Amaterasu canonically, but in theory this should be possible.