Neuromorphic computing could be used to directly perform complex classification tasks in hardware and is of potential value in the development of wearable, implantable and point-of-care devices. Successful implementation requires low-power operation, simple sensor integration and straightforward training. Organic materials are possible building blocks for neuromorphic systems, offering low-voltage operation and excellent tunability. However, systems developed so far still rely on external training in software. Here we report a neuromorphic biosensing platform that is capable of on-chip learning and classification. The modular biosensor consists of a sensor input layer, an integrated array of organic neuromorphic devices that form the synaptic weights of a hardware neural network and an output classification layer. We use the system to classify the genetic disease cystic fibrosis from modified donor sweat using ion-selective sensors; on-chip training is done using error signal feedback to modulate the conductance of the organic neuromorphic devices. We also show that the neuromorphic biosensor can be retrained on the chip, by switching the sensor input signals and alternatively through the formation of logic gates.