Maybe not to the extent where Arrival shows (you start seeing the future), but indeed it does.
I read this article years ago, comparing German and English bilinguals.
English has a grammatical toolkit for situating actions in time. German doesn’t have this feature. This linguistic difference seems to influence how speakers of the two languages view events. German speakers matched ambiguous scenes with goal-oriented scenes about 40% of the time on average, compared with 25% among English speakers. This difference implies that German speakers are more likely to focus on possible outcomes of people’s actions, but English speakers pay more attention to the action itself.
There are many other articles on the topic, like this and this, but it seems to be accepted that language does shape how we think. I don't think the field is mature enough to draw conclusive proof, but it seems to be plausible at this stage.
PS: My sister, a speech therapist, also tells me that language is in fact fundamental to how we think. She works with disabled kids (e.g., austists) whose behavior can be seen to change as specific communication patterns are taught to them.