POWER OF ATTORNEY

        Know all by these presents, that the undersigned hereby constitutes and
appoints each of Christian Vasquez and Richard Waldron, signing singly, the
undersigned's true and lawful attorney-in-fact to:

        (1) prepare, execute in the undersigned's name and on the undersigned's
            behalf, and submit to the U.S. Securities and Exchange Commission
            (the "SEC") a Form ID, including amendments thereto, and any other
            documents necessary or appropriate to obtain codes and passwords
            enabling the undersigned to make electronic filings with the SEC of
            reports required by Section 16(a) of the Securities Exchange Act of
            1934 or any rule or regulation of the SEC;

        (2) execute for and on behalf of the undersigned, in the undersigned's
            capacity as an officer and/or director of BioAtla, Inc. (the
            "Company"), Forms 3, 4, and 5 in accordance with Section 16(a) of
            the Securities Exchange Act of 1934 and the rules thereunder, and
            any other forms or reports the undersigned may be required to file
            in connection with the undersigned's ownership, acquisition, or
            disposition of securities of the Company;

        (3) do and perform any and all acts for and on behalf of the undersigned
            which may be necessary or desirable to complete and execute any such
            Form 3, 4, or 5, or other form or report, and timely file such form
            or report with the United States Securities and Exchange Commission
            and any stock exchange or similar authority; and

        (4) take any other action of any type whatsoever in connection with the
            foregoing which, in the opinion of such attorney-in-fact, may be of
            benefit to, in the best interest of, or legally required by, the
            undersigned, it being understood that the documents executed by such
            attorney-in-fact on behalf of the undersigned pursuant to this Power
            of Attorney shall be in such form and shall contain such terms and
            conditions as such attorney-in-fact may approve in such attorney-in-
            fact's discretion.

        The undersigned hereby grants to each such attorney-in-fact full power
and authority to do and perform any and every act and thing whatsoever
requisite, necessary, or proper to be done in the exercise of any of the rights
and powers herein granted, as fully to all intents and purposes as the
undersigned might or could do if personally present, with full power of
substitution or revocation, hereby ratifying and confirming all that such
attorney-in-fact, or such attorney-in-fact's substitute or substitutes, shall
lawfully do or cause to be done by virtue of this power of attorney and the
rights and powers herein granted. The undersigned acknowledges that the
foregoing attorneys-in-fact, in serving in such capacity at the request of the
undersigned, are not assuming, nor is the Company assuming, any of the
undersigned's responsibilities to comply with Section 16 of the Securities
Exchange Act of 1934.

        This Power of Attorney shall remain in full force and effect until the
undersigned is no longer required to file Forms 3, 4, and 5 with respect to the
undersigned's holdings of and transactions in securities issued by the Company,
unless earlier revoked by the undersigned in a signed writing delivered to the
foregoing attorneys-in-fact.

        IN WITNESS WHEREOF, the undersigned has caused this Power of Attorney to
be executed as of this 15th day of December, 2020.

                                        /s/ Mary Ann Gray
                                        ----------------------------------------
                                        Mary Ann Gray