According to the iPhone User Guide, this requires a third-party app with Files integration support. If you don't want to install one, it's actually quite easy to make one yourself.
- Install Xcode if you don't have it already. You'll need version 9 or above.
- Sign in using your Apple ID in the preferences. You have to do this in order to run your apps on an actual device.
- Create a new project and select the "Document Based App" template in the iOS tab. Name it "Files Local Storage" or something.
- Connect your device using a USB cable.
- Install it on your device! Select your iPhone out of the list in the toolbar, and click the big "play" button in the top left.
It won't have an icon, unless you go to the trouble of adding one. You'll also need to go into Settings and switch the storage location from "iCloud Drive" to "On My iPhone".
Also, unless you have a Developer ID, which costs $99/yr at the time of writing, you may need to reinstall your app every week or so.
If you don't want to do this, I've heard that Documents by Readdle supports Files integration. I've never tried it, and I'm not absolutely sure that it does.
On the other hand, I have tried GoodReader.
It doesn't support Files integration. (update: now it does!) It is also fantastic for file management, and has some good PDF features as well. It can even zip/unzip files and connect to various types of network servers, like SFTP. It's $4.99 (with 4.5 stars and 1,458 reviews) at the time of writing. Please be aware that it does have a built-in web browser.
(I don't have any affiliation with either of these apps.)