Yup! Well, technically, the Ahmadiyya Muslim community established a caliphate prior to end of the Ottomman caliphate, and continued it, instating a caliph as lately as 2003. The Wikipedia on them is weirdly extensive and it's fairly hard to find unbiased information about them. But they have also always been small and (to my knowledge) fringe and so do not get very much press.

An important note of context is that the caliphate is the claim that a single political entity represents the successor to Muhammad. There are Grand Imams and Grand Iyatollahs and the like who have no superior (in the absence of a caliph) but have made no attempt to use the term caliph.

Not quite the same, but a few people have claimed to be the promised Mahdi (the forerunner of the Messiah--think the John the Baptist) since the Caliphate's responsibilities was absorbed by the Türkiye Büyük Millet Meclisi (Grand National Assembly of Turkey--the Turkish parliament) in 1924. Hussein bin Ali, the then Hashemite Sharif of Mecca, claimed the title almost immediately, but was soon defeated by the House of Saud. Since then, the most successful claimant to the titles of either Mahdi or Caliphate was probably Muḥammad bin abd Allah al-Qahtani who seized the Grand Mosque in Mecca (Al-Masjid al-Haram) in 1979. While I don't think he ever called himself "Caliph", by being called "Mahdi" he was essentially a step above the Caliph. It didn't work out for him, in the end.