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The present study introduces a new family of analytic functions by utilizing the <i>q</i>-derivative operator and the <i>q</i>-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings.