  We consider one-dimensional stochastic Volterra equations with jumps for
which we establish conditions upon the convolution kernel and coefficients for
the strong existence and pathwise uniqueness of a non-negative c\`adl\`ag
solution. By using the approach recently developed in arXiv:2302.07758, we show
the strong existence by using a nonnegative approximation of the equation whose
convergence is proved via a variant of the Yamada--Watanabe approximation
technique. We apply our results to L\'evy-driven stochastic Volterra equations.
In particular, we are able to define a Volterra extension of the so-called
alpha-stable Cox--Ingersoll--Ross process, which is especially used for
applications in Mathematical Finance.
