Are there deterministic private-key encryption schemes that are CPA-secure? I'm assuming there can't be such schemes because CPA-security is equivalent to CPA-security for multiple encryptions, and
an adversary can distinguish between $(\mathsf{Enc}_k(m_0),\mathsf{Enc}_k(m_0))$ and $(\mathsf{Enc}_k(m_0),\mathsf{Enc}_k(m_1))$ if Enc is deterministic, hence there is no deterministic symmetric encryption scheme that is CPA-secure.
However, I can't see how an adversary might distinguish only one encryption from another.

The CPA indistinguishability experiment $\mathsf{PrivK}_{\mathcal A,\Pi}^{\mathsf{cpa}}(n)$:
Let $\Pi = (\mathsf{Gen}, \mathsf{Enc}, \mathsf{Dec})$ be any encryption scheme, $\mathcal A$ any adversary, and $n$ the security parameter.

A key $k$ is generated by running $\mathsf{Gen}(1^n)$.
The adversary $\mathcal A$ is given input $1^n$ and oracle access to $\mathsf{Enc}_k(\cdot)$, and outputs a pair of messages $m_0$, $m_1$ of the same length.
A uniform bit $b \in \{0, 1\}$ is chosen, and then a ciphertext $c \leftarrow \mathsf{Enc}_k(m_b)$ is computed and given to $\mathsf A$.
The adversary $\mathcal A$ continues to have oracle access to $\mathsf{Enc}_k(\cdot)$, and outputs a bit $b'$.
The experiment is defined to be $1$ if $b' = b$, and $0$ otherwise.

Definition of CPA-security:
A private-key encryption scheme $\Pi = (\mathsf{Gen}, \mathsf{Enc}, \mathsf{Dec})$ has indistinguishable encryptions under a chosen-plaintext attack, or is CPA-secure, if for all probabilistic polynomial-time adversaries $\mathcal A$ there is a negligible function $\mathsf{negl}$ such that
$$\text{Pr}[\mathsf{PrivK}_{\mathcal A,\Pi}^{\mathsf{cpa}}(n) = 1] \leq \frac{1}{2} + \mathsf{negl}(n),$$
where the probability is taken over the randomness used by $\mathcal A$, as well as the randomness used in the experiment.
Summarizing fkraiem's comment, a CPA-secure encryption scheme can not be deterministic.

The reason is simple: the attacker is challenged to distinguish between the encryption of $m_0$ and $m_1$, but he also has access to an encryption oracle (and he can query the oracle at whatever input he wants!). If the encryption scheme yields the same ciphertexts each time it's queried on the same plaintexts, the attacker can simply ask the oracle for an encryption of $m_0$ and an encryption of $m_1$, and compare which one is equal to the given challenge ciphertext.