SECTION 16 POWER OF ATTORNEY I, Gary L. Hinkle, do hereby constitute and appoint Roy E. Halyama, Kirtan Parikh, Matthew Rucker, and Daniel Kendle my true and lawful attorneys-in-fact, any of whom acting singly is hereby authorized, for me and in my name and on my behalf as a director, officer and/or shareholder of Burke & Herbert Financial Services Corp., to (i) prepare, execute in my name and on my behalf, and submit to the U.S. Securities and Exchange Commission (the "SEC") a Form ID, including any necessary amendments thereto, and any other documents necessary or appropriate to obtain or update codes and passwords enabling me to make electronic filings with the SEC ofreports required by Section 16(a) of the Securities Exchange Act of 1934 or any rule or regulation of the SEC in respect thereof; and (ii) prepare, execute and file any and all forms, instruments or documents, including any necessary amendments thereto, as such attorneys or attorney deems necessary or advisable to enable me to comply with Section 16 of the Securities Exchange Act of 1934 or any rule or regulation of the SEC in respect thereof (collectively, "Section 16"). I do hereby ratify and confirm all acts my said attorney shall do or cause to be done by virtue hereof. I acknowledge that the foregoing attorneys-in-fact, serving in such capacity at my request, are not assuming, nor is Burke & Herbert Financial Services Corp. assuming, any of my responsibilities to comply with Section 16. This power of attorney shall remain in full force and effect until it is revoked by the undersigned in a signed writing delivered to each such attorney-in-fact or the undersigned is no longer required to comply with Section 16, whichever occurs first. WITNESS the execution hereof this 3rd day of May, 2024. 170234503vl