How to contruct a pseudorandom with a specific property
Depending on what you consider an acceptable approach, you could get there by having a random-looking function from your PRF domain to a prime-order cyclic group, and then use the permutation $x \mapsto x^k$ on the group.

Under reasonable assumptions and choice of group, this is a PRF. And since the key-dependent part is a permutation, two keys will never map the same $i$ to the same function value.

So this construction satisfies your requirements, but it is not a tribe ensemble.