Flatworms, such as planaria, absolutely sleep and have a daily rhythm of increased and decreased activity. Cnidarians, such as hydra or the cassiopea jellyfish, sleep as well. Flatworms have a CNS and a bilobed brain, though cnidarians do not. My wife worked extensively in a planaria lab (one mentioned in citation below, coincidentally) before getting her PhD. Sponges (which lack any neurons at all) have a diurnal rhythm and the sleep research field considers it an open question whether they sleep, with one opined hypothesis being that sleep evolved for gut regulation before it was adapted for nervous regulation. Some go so far as to suggest the possibility that bacteria, algae, and plants sleep.
Neurotransmitters of sleep and wakefulness in flatworms https://pmc.ncbi.nlm.nih.gov/articles/PMC9216492/
"Sleep is a prominent behavioral and biochemical state observed in all animals studied, including platyhelminth flatworms."
"Dopamine and histamine decreased the time spent inactive and increased distance traveled, consistent with their wake-promoting effect in vertebrates and fruit flies; pyrilamine increased restfulness and GABA showed a nonsignificant trend towards promoting restfulness in a dose-dependent manner, in agreement with their sleep-inducing effect in vertebrates, fruit flies, and Hydra."
Persistence of Nocturnality in Decapitated and Bisected Flatworms https://pmc.ncbi.nlm.nih.gov/articles/PMC10278384/
"Here, we demonstrate that intact flatworms were predominantly active at night, with peaks in activity seen in the hours after lights-off and before lights-on. While decapitated and tailless flatworms could still move, both were less active than the original animal, and both segments retained a nocturnal lifestyle. Furthermore, decapitated flatworms, once regenerated, again showed a U-shaped pattern of nocturnal activity reminiscent of the two night-time peaks seen in the original animal. These results could be used to further investigate how regeneration may affect motor control and motor output, or to further investigate the presence of a clock in the flatworm brain."
"Nocturnal by nature, their rhythm persists even in the absence of photoperiodic cues, suggestive of an endogenous circadian clock (Omond et al., 2017). Even more interestingly, despite being closely related to other lophotrochozoans, that is, annelid (segmented) worms and mollusks (Tessmar-Raible, 2003), flatworms have secondarily lost their circulatory and respiratory systems, and endocrine glands. What remains is (1) a centralized nervous system, complete with bilobed brain (Roberts-Galbraith and Newmark, 2015); (2) a need for sleep that is regulated by sleep-wake history and induced by the evolutionarily conserved neurotransmitter gamma-aminobutyric acid, or GABA;"
A sleep-like state in Hydra unravels conserved sleep mechanisms during the evolutionary development of the central nervous system https://www.science.org/doi/10.1126/sciadv.abb9415
"Hydra sleep was shaped by homeostasis and necessary for cell proliferation, but it lacked free-running circadian rhythms. Instead, we detected 4-hour rhythms that might be generated by ultradian oscillators underlying Hydra sleep. Microarray analysis in sleep-deprived Hydra revealed sleep-dependent expression of 212 genes, including cGMP-dependent protein kinase 1 (PRKG1) and ornithine aminotransferase. Sleep-promoting effects of melatonin, GABA, and PRKG1 were conserved in Hydra. However, arousing dopamine unexpectedly induced Hydra sleep. Opposing effects of ornithine metabolism on sleep were also evident between Hydra and Drosophila, suggesting the evolutionary switch of their sleep-regulatory functions. Thus, sleep-relevant physiology and sleep-regulatory components may have already been acquired at molecular levels in a brain-less metazoan phylum and reprogrammed accordingly."
The Jellyfish Cassiopea Exhibits a Sleep-like State
https://www.cell.com/current-biology/fulltext/S0960-9822(17)...
"In Cnidaria, neurons are organized into a non-centralized radially symmetric nerve net [11, 13, 15–17] that nevertheless shares fundamental properties with the vertebrate nervous system: action potentials, synaptic transmission, neuropeptides, and neurotransmitters [15–20]. It was reported that cnidarian soft corals [21] and box jellyfish [22, 23] exhibit periods of quiescence, a pre-requisite for sleep-like states, prompting us to ask whether sleep is present in Cnidaria. Within Cnidaria, the upside-down jellyfish Cassiopea spp. displays a quantifiable pulsing behavior, allowing us to perform long-term behavioral tracking. Monitoring of Cassiopea pulsing activity for consecutive days and nights revealed behavioral quiescence at night that is rapidly reversible, as well as a delayed response to stimulation in the quiescent state. When deprived of nighttime quiescence, Cassiopea exhibited decreased activity and reduced responsiveness to a sensory stimulus during the subsequent day, consistent with homeostatic regulation of the quiescent state. Together, these results indicate that Cassiopea has a sleep-like state, supporting the hypothesis that sleep arose early in the metazoan lineage, prior to the emergence of a centralized nervous system."
OPINION - Exploring phylogeny to find the function of sleep https://www.nature.com/articles/s41583-018-0098-9
"Indeed, it is interesting to consider sleep in animals that lack neurons altogether, such as sponges and the Placozoan species Trichoplax adhaerens. T. adhaerens have gland cells that are likely to contain and secrete neuropeptides91. Despite lacking neurons and muscle, T. adhaerens sense and respond to their environment and move and eat via the coordinated action of their cilia92,93,94. Sponges also lack muscles and neurons but carry genes encoding synaptic scaffold proteins95, can contract coordinately with a diurnal rhythm96 and can respond to their environment97. Evidence that Placozoa spp. or Porifera spp. have a sleep state would demonstrate that sleep is not just for organisms with neurons and would also suggest that non-neuronal cells can organize sleep behaviour."
"Many plants synthesize melatonin99, although its function in plants is not elucidated100. Cassiopea jellyfish species live in a symbiotic relationship with single-cell photosynthetic algae, which provide carbohydrate fuel to these jellyfish. Do these algae also ‘sleep’ at night when photosynthetic activity is absent? Mammals are colonized with intestinal microbiota, the composition of which changes with sleep deprivation101. Should these microorganisms be considered to be sleepers? If, at its core, sleep were serving a metabolic function (see below), it is not inconceivable that plants, algae and single-cell prokaryotes will also ultimately be considered to sleep."