Local rate control is a key enabler to generalize image and video compression for dedicated challenges, such as video coding for machines. While traditional hybrid video coding can easily adapt the local rate-distortion trade-off by changing the local quantization parameter, no such approach is currently available for learning-based video compression. In this paper, we propose LRC-DHVC, a hierarchical video compression network, which allows continuous local rate control on a pixel level to vary the spatial quality distribution within individual video frames. This is achieved by concatenating a quality map to the input frame and applying a weighted MSE loss which matches the pixelwise trade-off factors in the quality map. During training, the model sees a variety of quality maps due to a constrained-random generation. Our model is the first neural video compression network, which can continuously and spatially adapt to varying quality constraints. Due to the wide quality and bit rate range, a single set of network parameters is sufficient. Compared to single rate point networks, which scale linearly with the number of rate points, the memory requirements for our network parameters remain constant. The code and model are available at link-updated-upon-acceptance.