Possibly the nearest to applying capabilities to C is pointer authentication: https://lwn.net/Articles/718888/
(This is one of those times where the C memory model as described in the spec is very different from the mental PDP-11 that C programmers actually use to reason about)
C's file descriptors are capabilities.
CheriBSD is definitely closer: https://www.cheribsd.org/. Or one of the other CHERI projects.
C is not really dependent on a flat memory model - instead, it models memory allocations as separate "objects" (quite reniniscent of "object orientation in hardware" which is yet another name for capabilities), and a pointer to "object" A cannot be offset to point into some distinct "object" B.
> A Truly Relational Language
This is broadly speaking how PROLOG and other logic-programming languages work. The foundational operation in such languages is a knowledge-base query, and "relations" are the unifying concept as opposed to functions with predefined inputs and outputs.