$UP^{\ O}\neq P^{\ O}$ for some oracle $O$
Here is the construction from Rackoff's paper Relativized questions involving probabilistic automata. We construct the oracle $O$ using diagonalization, in steps. At step $t$ we will construct the initial segment $O_t$ of $O$ which consists of $O \cap \bigcup_{i=0}^{n_t} \Sigma^i$ for some $n_t$, where $n_0 &lt; n_1 &lt; \cdots$ is an increasing sequence. We will maintain the following invariant: $O_t$ contains at most one string of each length. This will guarantee that the following language is in $UP^O$:
$$
L = \{ x : \exists y \in O \cap \Sigma^{|x|} \}.
$$

The construction will use an enumeration $M_i$ of all polynomial time Turing machines, with the guarantee that the running time of $M_i$ is at most $in^i$. It is a standard fact that such an enumeration exists.

Let $O_{t-1}$ be the initial segment constructed at step $t-1$ (or $\emptyset$ in the first step). We now show how to construct $O_t$. Choose the smallest $a_t &gt; n_{t-1}$ such that $ta_t^t &lt; 2^{a_t}$. Run $M_t$ on the string $0^{a_t}$, answering NO to all queries outside of $O_{t-1}$. Let $b_t$ be the length of the largest string queried, and let $n_t = \max(a_t,b_t)$. If $M_t$ accepts, set $O_t = O_{t-1}$, guaranteeing that $0^{a_t} \notin L$. Otherwise, set $O_t = O_{t-1} \cup \{x\}$, where $x$ is some string of length $a_t$ which $M_t$ didn't query (such a string must exist since $ta_t^t &lt; 2^{a_t}$), guaranteeing that $0^{a_t} \in L$. You can check that $O_t$ satisfies the invariant stated above.

By construction, $L \in UP^O$ (the UP machine guesses $y$ of the same length as the input and verifies that $y \in O$), while $L \notin P^O$ since $M_t$ answers incorrectly on the string $0^{a_t}$, by construction.

Beigel, in the paper On the relativized power of additional accepting paths, shows that $P^O \neq UP^O$ for a random oracle, crediting this result to Rudich's PhD thesis. In fact he proves even stronger separations, showing that allowing more accepting paths increases the (relativized) power of polytime machines.