Can we develop a model that can synthesize realistic speech directly from a latent space, without explicit conditioning? Despite several efforts over the last decade, previous adversarial and diffusion-based approaches still struggle to achieve this, even on small-vocabulary datasets. To address this, we propose AudioStyleGAN (ASGAN) -- a generative adversarial network for unconditional speech synthesis tailored to learn a disentangled latent space. Building upon the StyleGAN family of image synthesis models, ASGAN maps sampled noise to a disentangled latent vector which is then mapped to a sequence of audio features so that signal aliasing is suppressed at every layer. To successfully train ASGAN, we introduce a number of new techniques, including a modification to adaptive discriminator augmentation which probabilistically skips discriminator updates. We apply it on the small-vocabulary Google Speech Commands digits dataset, where it achieves state-of-the-art results in unconditional speech synthesis. It is also substantially faster than existing top-performing diffusion models. We confirm that ASGAN's latent space is disentangled: we demonstrate how simple linear operations in the space can be used to perform several tasks unseen during training. Specifically, we perform evaluations in voice conversion, speech enhancement, speaker verification, and keyword classification. Our work indicates that GANs are still highly competitive in the unconditional speech synthesis landscape, and that disentangled latent spaces can be used to aid generalization to unseen tasks. Code, models, samples: https://github.com/RF5/simple-asgan/