Split vector layer a get a separate layer for each district (polygon feature).

Run Clip Raster by Mask Layer in Batch Mode. For the output, define a path and when promted, for Autofill mode select Fill with parameter values > Mask Layer - or, to get more sophisticated options how to name the files based on attribute values, use Autofill... > Calculate by Expression...

Split vector layer to get a separate layer for each district (polygon feature).

Run Clip Raster by Mask Layer in Batch Mode. For the output, define a path and when promted, for Autofill mode select Fill with parameter values > Mask Layer - or, to get more sophisticated options how to name the files based on attribute values, use Autofill... > Calculate by Expression...