This paper addresses issues concerning biometric authentication based on handwritten signatures. Our research aimed to check whether a handwritten signature acquired with a mobile device can effectively verify a user's identity. We present a novel online signature verification method using coordinates of points and pressure values at each point collected with a mobile device. Convolutional neural networks are used for signature verification. In this paper, three neural network models are investigated, i.e., two self-made light SigNet and SigNetExt models and the VGG-16 model commonly used in image processing. The convolutional neural networks aim to determine whether the acquired signature sample matches the class declared by the signer. Thus, the scenario of closed set verification is performed. The effectiveness of our method was tested on signatures acquired with mobile phones. We used the subset of the multimodal database, MobiBits, that was captured using a custom-made application and consists of samples acquired from 53 people of diverse ages. The experimental results on accurate data demonstrate that developed architectures of deep neural networks can be successfully used for online handwritten signature verification. We achieved an equal error rate (EER) of 0.63% for random forgeries and 6.66% for skilled forgeries.