Communication complexity problems with linear distance
Let $C:\{0,1\}^{n} \to \{0,1\}^{2n}$ be an error correcting code with linear distance. Let $g: \{0,1\}^{n} \times \{0,1\}^{n} \to \{0,1\}$ be a function whose randomized communication complexity is large (say, $\Omega(\sqrt{n})$ or $\Omega(n))$.

Define $f: \{0,1\}^{2n} \times \{0,1\}^{2n} \to \{0,1,*\}$ to be the partial function that on codewords of $C$ outputs $f(x,y) = g(C^{-1}(x),C^{-1}(y))$, and it outputs $*$ if at least one of $x,y$ is not in $C$.

Clearly, the communication complexity of $f$ is equal to the communication complexity of $g$, and $f$ satisfies the property that for every two different inputs on which $f$ outputs 0 or 1, the distance between them is linear.