We report on a soft x-ray microscope using a gas-discharge plasma with pseudo spark-like electrode geometry as a light source. The source produces a radiant intensity of $4\times {10}^{13}\text{\hspace{0.17em} photons}∕\left(\mathrm{sr}\text{\hspace{0.17em} pulse}\right)$ for the $2.88\mathrm{nm}$ emission line of helium-like nitrogen. At a demonstrated $1\mathrm{kHz}$ repetition rate a brilliance of $4.3\times {10}^9\text{\hspace{0.17em} photons}∕\left(\mu {\mathrm{m}}^2\mathrm{sr}\mathrm{s}\right)$ is obtained for the $2.88\mathrm{nm}$ line. Ray-tracing simulations show that, employing an adequate grazing incidence collector, a photon flux of $1\times {10}^7\text{\hspace{0.17em} photons}∕\left(\mu {\mathrm{m}}^2\mathrm{s}\right)$ can be achieved with the current source. The applicability of the presented pinch plasma concept to soft x-ray microscopy is demonstrated in a proof-of-principle experiment.