Applications in which researchers aim to extract a single land type from remotely sensed data are quite common in practical scenarios: extract the urban footprint to make connections with socio-economic factors; map the forest extent to subsequently retrieve biophysical variables and detect a particular crop type to successively calibrate and deploy yield prediction models. In this scenario, the (positive) targeted class is well defined, while the negative class is difficult to describe. This one-class classification setting is also referred to as positive unlabelled learning (PUL) in the general field of machine learning. To deal with this challenging setting, when satellite image time series data are available, we propose a new framework named positive and unlabelled learning of satellite image time series (PUL-SITS). PUL-SITS involves two different stages: In the first one, a recurrent neural network autoencoder is trained to reconstruct only positive samples with the aim to higight reliable negative ones. In the second stage, both labelled and unlabelled samples are exploited in a semi-supervised manner to build the final binary classification model. To assess the quality of our approach, experiments were carried out on a real-world benchmark, namely Haute-Garonne, located in the southwest area of France. From this study site, we considered two different scenarios: a first one in which the process has the objective to map Cereals/Oilseeds cover versus the rest of the land cover classes and a second one in which the class of interest is the Forest land cover. The evaluation was carried out by comparing the proposed approach with recent competitors to deal with the considered positive and unlabelled learning scenarios.