Short answer. Yes. Article 1358 requires a public document for any power to administer property, or any other power, whose object is an act appearing or that should appear in a public document. Since the acts you are authorizing must themselves be in a public document, the power of attorney must be one too.
What the law says
The power to administer property, or any other power which has for its object an act appearing or which should appear in a public document, or should prejudice a third person
Civil Code, Article 1358 — What Must Be in a Public Document. Read the full provision →
The power of attorney follows the form of the act it authorizes
Article 1358 lists among the acts that must appear in a public document: the power to administer property, or any other power which has for their object an act appearing or which should appear in a public document, or should prejudice a third person. This provision ties the required form of the power of attorney directly to the required form of the underlying act. If the act you are authorizing, such as selling real property or transmitting real rights, must itself appear in a public document, the power authorizing your attorney-in-fact to perform that act is swept into the same requirement.
Why a private document falls short here
A private, unnotarized power of attorney might be sufficient for many ordinary tasks, but it does not carry the same formal weight as a public document that is executed before a notary public and entered in the notarial register. Since the acts your attorney-in-fact will perform on your behalf need to appear in a public document, granting that authority through a merely private instrument creates a mismatch between the form of the authorization and the form the underlying transaction legally requires.
The provision also reaches acts that would prejudice third persons
Article 1358 extends the public document requirement to a power whose object should prejudice a third person, in addition to powers tied to acts that appear or should appear in a public document. This broadens the coverage beyond just real property transactions, so it is worth checking whether the specific authority you are granting could affect a third party's rights, since that alone can independently trigger the same public document requirement regardless of whether real property is directly involved.
What to do before executing the power of attorney
Given that the acts your attorney-in-fact will undertake must appear in a public document, you should execute the power of attorney itself as a public document, properly notarized, rather than as a private writing. Doing this before the attorney-in-fact attempts to act on your behalf avoids the risk that the underlying transaction, or the authority behind it, will be challenged later for failing to meet the form the law requires for acts of this kind.