Is it true that some small cars (i.e. "subcompact") can be denied by Lyft despite them being four-doors and meeting all requirements?

Yes, new rule starting 2/25/2019. You can be grandfathered, if you already owned a car.

Yes, you heard correctly. Starting 2/25/2019, Lyft now prohibits certain "subcompact" vehicles and have published a list for "Ineligible makes and models" in addition to their regular vehicle requirements.

The idea is that they prefer larger cars to satisfy their requirements for their standard Lyft vehicles, and they deemed certain cars to be too small.  These include cars like Mini Cooper, Hyundai Accent, Volkswagen Beetle, BMW i3, etc.

Here's their statement:

To provide a comfortable rider experience, certain subcompact vehicles are not eligible on the Lyft platform as of February 25, 2019. Subcompact vehicles typically have an interior combined passenger and cargo volume index of less than 100 cubic feet, which makes for a tight space for passengers.

See a list of ineligible subcompact vehicles.

If you're an existing driver and you were approved to drive with a subcompact vehicle prior to February 25, 2019, you'll be able to continue driving with your existing subcompact vehicle. You won't be able to add a second vehicle that is classified as subcompact, or replace your existing vehicle with a different subcompact vehicle.

https://s3.amazonaws.com/lyft-assets/helpcenter/Policies/subcompact_vehicle_requirements.pdf

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC