Attention deficit hyperactivity disorder (ADHD) is believed to be connected with a high level of hyperactivity caused by alterations of the control of dopaminergic transmission in the brain. The strain of hyperdopaminergic dopamine transporter knockout (DAT-KO) rats represents an optimal model for investigating ADHD-related pathological mechanisms. The goal of this work was to study the influence of the overactivated dopamine system in the brain on a motor cognitive task fulfillment. The DAT-KO rats were trained to learn an object recognition task and store it in long-term memory. We found that DAT-KO rats can learn to move an object and retrieve food from the rewarded familiar objects and not to move the non-rewarded novel objects. However, we observed that the time of task performance and the distances traveled were significantly increased in DAT-KO rats in comparison with wild-type controls. Both groups of rats explored the novel objects longer than the familiar cubes. However, unlike controls, DAT-KO rats explored novel objects significantly longer and with fewer errors, since they preferred not to move the non-rewarded novel objects. After a 3 months’ interval that followed the training period, they were able to retain the learned skills in memory and to efficiently retrieve them. The data obtained indicate that DAT-KO rats have a deficiency in learning the cognitive task, but their hyperactivity does not prevent the ability to learn a non-spatial cognitive task under the presentation of novel stimuli. The longer exploration of novel objects during training may ensure persistent learning of the task paradigm. These findings may serve as a base for developing new ADHD learning paradigms.