Eye-tracking technology is an innovative tool that holds promise for enhancing dementia screening. In this work, we introduce a novel way of extracting salient features directly from the raw eye-tracking data of a mixed sample of dementia patients during a novel instruction-less cognitive test. Our approach is based on self-supervised representation learning where, by training initially a deep neural network to solve a pretext task using well-defined available labels (e.g. recognising distinct cognitive activities in healthy individuals), the network encodes high-level semantic information which is useful for solving other problems of interest (e.g. dementia classification). Inspired by previous work in explainable AI, we use the Layer-wise Relevance Propagation (LRP) technique to describe our network's decisions in differentiating between the distinct cognitive activities. The extent to which eye-tracking features of dementia patients deviate from healthy behaviour is then explored, followed by a comparison between self-supervised and handcrafted representations on discriminating between participants with and without dementia. Our findings not only reveal novel self-supervised learning features that are more sensitive than handcrafted features in detecting performance differences between participants with and without dementia across a variety of tasks, but also validate that instruction-less eye-tracking tests can detect oculomotor biomarkers of dementia-related cognitive dysfunction. This work highlights the contribution of self-supervised representation learning techniques in biomedical applications where the small number of patients, the non-homogenous presentations of the disease and the complexity of the setting can be a challenge using state-of-the-art feature extraction methods.