So far it looks like
polkadot.js/api supports ECDSA signatures, but the polkadot.js extension doesn't expose a method to use ECDSA:
It doesn't seem to be so, however there's a tracking bug for it. The moonbase docs recommend using the in-browser local storage. If you try to submit the extrinsic with an ecdsa signature using the extension over the moonbase network (which is using ecdsa as a signature method), then it will fail with a mismatch of types.