Claims

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1964_03_01_erdos: Erdős (Michigan Math. J., 1964) proves that when some cardinal lies strictly between m and the continuum, analytic functions taking at most m values at each point number at most m; formalized by others.

2017_01_05_kumar_shelah: Shows that after adding aleph one Cohen reals to a model with continuum aleph two, every continuum-sized family of entire functions takes continuum many values at some point, so a yes answer is consistent; refereed in 2017.

2023_10_30_schilhan_weinert: Forces, over a model of GCH, aleph two entire functions taking at most aleph one values at each point while the continuum is aleph two, so a no answer is consistent; with Kumar and Shelah's model, independence of ZFC.

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