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The distinctive effects of fuzzy dark matter are most visible at non-linear galactic scales. We present the first simulations of mixed fuzzy and cold dark matter, obtained with an extended version of the Nyx code. Fuzzy (or ultralight, or axion-like) dark matter dynamics are governed by the comoving Schrödinger-Poisson equation. This is evolved with a pseudospectral algorithm on the root grid, and with finite differencing at up to six levels of adaptive refinement. Cold dark matter is evolved with the existing N-body implementation in Nyx. We present the first investigations of spherical collapse in mixed dark matter models, focusing on radial density profiles, velocity spectra and soliton formation in collapsed halos. We find that the effective granule masses decrease in proportion to the fraction of fuzzy dark matter which quadratically suppresses soliton growth, and that a central soliton only forms if the fuzzy dark matter fraction is greater than 10\%. The Nyx framework supports baryonic physics and key astrophysical processes such as star formation. Consequently, AxioNyx will enable increasingly realistic studies of fuzzy dark matter astrophysics.