This Gist shows how the Volley Request<T> class can be extended to support file download. The response listener will receive a File through the callback method onResponse(File downloadedFile). The file will already be on the device at this point.

FileDownloadRequest.java

package za.org.pta.gdg.gists;

import java.io.File;
import java.io.FileOutputStream;
import java.util.Date;

import android.content.Context;
import android.os.Environment;
import android.util.Log;

import com.android.volley.NetworkResponse;
import com.android.volley.Request;
import com.android.volley.Response;
import com.android.volley.Response.ErrorListener;
import com.android.volley.VolleyError;

/**
 * This is a File Download request using the Volley library in Android. 
 * Essentially it parses network response and then writes this into a File object and return this to the response listener.
 * 
 * @author Ishmael Makitla, 2015
           Country Mentor: Google Developer Groups South Africa,
	   GDG-Pretoria
 *
 */
public class FileDownloadRequest extends Request<File>{
	private static final String TAG = FileDownloadRequest.class.getSimpleName();
	//the listener of our custom File Download Request expects a File
	private  Response.Listener<File> mReponseListener;
	private ErrorListener mErrorListener;
	//this is the name of the file the user wants to download 
	//- so it is the last segment of the HPPT-GET URL (www.mystuff.com/downloads/file1.json) = file1.json
    private String targetFileName;
    
    private Context context;
	
	public FileDownloadRequest(Context ctx, String url, ErrorListener errorListener) {
		super(Method.GET, url, errorListener);	
		mErrorListener = errorListener;
		targetFileName = getFileNameFromURL(url);
		context = ctx;
	}
	
	public FileDownloadRequest (Context ctx, String url, Response.ErrorListener errorListener, Response.Listener<File> listener){
		super(Method.GET, url, errorListener);	
		mReponseListener = listener;
		mErrorListener = errorListener;
		targetFileName = getFileNameFromURL(url);
		context = ctx;
	}
	
	private String getFileNameFromURL(String url){
		if(url == null || url.trim().isEmpty()){ return null; }
		String filename = null;
		try{
			filename = url.substring(url.lastIndexOf(File.separator)+1);
			Log.i(TAG, "Target-File-Name For URL "+ url+" is "+filename);
		}
		catch(Exception e){
			e.printStackTrace();
		}
		
		return filename;
	}

	@Override
	protected void deliverResponse(File downloadedFile) {
		
		if(downloadedFile != null){
			Log.i(TAG, "deliverResponse Downloaded File =  "+downloadedFile.getPath());
			mReponseListener.onResponse(downloadedFile);
		}
		else{
			//Not a file - error
			Log.i(TAG, "The content received from Server is not a valid file.  ");
			mErrorListener.onErrorResponse(new VolleyError("The content received from Server is not a valid file."));
		}		
	}

	@Override
	protected Response<File> parseNetworkResponse(NetworkResponse networkResponse) {
		Log.i(TAG, "parseNetworkResponse. Data = "+networkResponse.data);
		//write the response string into into a file (.json file in our case) and then pass onto the listener
		byte[] data = networkResponse.data;
		FileOutputStream fileOutput;
		File downloadedFile = null;
		try{	
			String filename = (targetFileName != null && !targetFileName.trim().isEmpty()? targetFileName : String.valueOf(new Date().getTime())+".json"); 
			Log.i(TAG, "parseNetworkResponse. File-Name = "+filename);
			//Ideally, you could take the download path as a parameter and that because the location of the downloaded file
			downloadedFile = new File(Environment.getExternalStorageDirectory()+File.separator+filename);
			downloadedFile.createNewFile();
			
			fileOutput = new FileOutputStream(downloadedFile); 			
			fileOutput.write(data);
			fileOutput.close();	
						
			Log.i(TAG, "parseNetworkResponse. new File(filename) = downloadedFile = "+downloadedFile.getName());
			
			//now we have the file, deliver it as success-response payload
			Log.i(TAG, "parseNetworkResponse. Before Response.success. downloadedFile = "+downloadedFile);
			return Response.success(downloadedFile, getCacheEntry());		
		}
		catch(Exception e){
			Log.e(TAG, "parseNetworkResponse. Error at this point", e);
			e.printStackTrace();
			return null;
		}
		
	}

FileDownloadUsageExample.java

//Here is a usage example for the FileDownloadRequest gists
 RequestQueue queue = Volley.newRequestQueue(context);
				  FileDownloadRequest fileDownloadRequest = new FileDownloadRequest(context,"http://my-file-server/files/file1.txt", new Response.ErrorListener() {
						@Override
						public void onErrorResponse(VolleyError error) {
							try {									
								Log.w(TAG, "There was an error downloading a file. Error is "+error);
							} 
							catch (NullPointerException err) {Log.e(TAG, err.getLocalizedMessage(), err);}
						}
				}, new Response.Listener<File>() {
					@Override
					public void onResponse(File file) {
						Log.i(TAG, "Got file: "+file); //we got the file file1.txt
						//do whatever with this file here - if it was a audio/video, you could start playing it
					}
				});					
				queue.add(fileDownloadRequest);

I added a usage example of the gist on separate file.