It's a homework question. I think that one cannot exist. My reasoning is as follows: Let (X, d) be totally bounded. Let $a_n$ be an infinite sequence of decreasing positive real numbers converging to $0$. For each $i$ let $A_i$ be a finite collection of open balls of radius $a_i$ that cover X. Let $P_i$ be the set of points the balls in $A_i$ are centered around. $A_i$ exists for each $i$ because X is totally bounded. The union of all $P_i$ is countable. Then for each $x$ in $X$ and for all $\epsilon > 0$ we can find a point $p$ in $\cup{P_i}$ with $d(x,p) < \epsilon$. Then $\cup{P_i}$ is a countable dense subset of X, so X is separable.
Am I right, or on the right track? I didn't use the fact that X is not complete.