Automatic speech recognition (ASR) of code-switching speech requires careful handling of unexpected language switches that may occur in a single utterance. In this paper, we investigate the feasibility of using multilingually trained deep neural networks (DNN) for the ASR of Frisian speech containing code-switches to Dutch with the aim of building a robust recognizer that can handle this phenomenon. For this purpose, we train several multilingual DNN models on Frisian and two closely related languages, namely English and Dutch, to compare the impact of single-step and two-step multilingual DNN training on the recognition and code-switching detection performance. We apply bilingual DNN retraining on both target languages by varying the amount of training data belonging to the higher-resourced target language (Dutch). The recognition results show that the multilingual DNN training scheme with an initial multilingual training step followed by bilingual retraining provides recognition performance comparable to an oracle baseline recognizer that can employ language-specific acoustic models. We further show that we can detect code-switches at the word level with an equal error rate of around 17% excluding the deletions due to ASR errors.