A pressure control device can include a releasable assembly and a latch including a piston having set and unset positions, and the releasable assembly including a release dog extendable into engagement with the latch. The piston can displace with the extended dog to the unset position. A method of retrieving a releasable assembly can include engaging a running tool with the releasable assembly, changing a fluid flow rate through the running tool, thereby securing the running tool to the releasable assembly, applying a force from the running tool to the releasable assembly, thereby extending a release dog from the releasable assembly, engaging the release dog with a latch of the pressure control device and deactivating the latch, and then retrieving the releasable assembly.