A broad range of studies demonstrate that sleep has a facilitating role in memory consolidation (see Rasch & Born, ). Whether sleep‐dependent memory consolidation is also apparent in infants in their first few months of life has not been investigated. We demonstrate that 3‐month‐old infants only remember a cartoon face approximately 1.5–2 hours after its first presentation when a period of sleep followed learning. Furthermore, habituation time, that is, the time to become bored with a stimulus shown repetitively, correlated negatively with the density of infant sleep spindles, implying that processing speed is linked to specific electroencephalographic components of sleep. Our findings show that without a short period of sleep infants have problems remembering a newly seen face, that sleep enhances memory consolidation from a very early age, highlighting the importance of napping in infancy, and that infant sleep spindles may be associated with some aspects of cognitive ability. Graphical AbstractThe contents of this page will be used as part of the graphical abstract of html only. It will not be published as part of main. We studied whether the memory of infants as young as three‐month old benefit from a short nap during the day. Infants were only able to remember the previously shown cartoon face when they had a nap after learning. Furthermore, specific electroencephalographic components of sleep, sleep spindles correlated with the time needed for becoming habituated to the new stimulus.