During task-oriented dialogues (TODs), human users naturally introduce chitchat that is beyond the immediate scope of the task, interfering with the flow of the conversation. To address this issue without the need for expensive manual data creation, we use few-shot prompting with Llama-2-70B to enhance the MultiWOZ dataset with user backstories, a typical example of chitchat interference in TODs. We assess the impact of this addition by testing two models: one trained solely on TODs and another trained on TODs with a preliminary chitchat interaction. Our analysis reveals that our enriched dataset poses a significant challenge to these systems. Moreover, we demonstrate that our dataset can be effectively used for training purposes, enabling a system to consistently acknowledge the user's backstory while also successfully moving the task forward in the same turn, as confirmed by human evaluation. These findings highlight the benefits of generating novel chitchat-TOD scenarios to test TOD systems more thoroughly and improve their resilience to natural user interferences.