For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial function of two or more variables?

Looking at the Lagrange Interpolation formula in the univariate case I try to argue for the 2-variable case this way.

So given some data $(a_i,b_j)\mapsto c_{ij}$ . I can try to fit a many polynomial functions $F_i$'s with $F_i(a_i,b_j)=c_{ij}$ for all $j $.  The coefficients of $F_i$'s vary as rational functions in the given data $c_{ij}$.
As inversion is also a polynomial function in one variable can  I assume I can get a polynomial of two variable to represent all functions of two variables?