This 1963 newspaper article has a photo of a mobile phone prototype

An article from the April 18, 1963 edition of the Massfield News-Journal promised "You'll be able to carry a phone in pocket in future."

Right now, it's a laboratory development and it's workable, allowing the carrier to make and answer calls wherever he may be.

Other telephones of the future includes a kitchen loud speaking telephone, and a visual image telephone.
The kitchen instrument can be used as regular telephone, a loudspeaking phone if the housewife happens to be busy preparing a meal, or as an intercom station for the home.

The visual image telephone allows the parties to converse by way of a microphone and loud speaker while a miniature television camera transmits the image. The "TV phone" also will have a writer signature transmission system and a conversation tape recorder.

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Related, a cartoon from the March 5th, 1919 edition of The Daily Mirror about the inconveniences of having a mobile phone:

And, going back even further, this excerpt from Ella Cheever Thayer's delightful 1979 melodrama novel, Wired Love: A Romance of Dots and Dashes:

We will soon be able to do everything by electricity; who knows but some genius will invent something for the especial use of lovers? Something, for instance, to carry in their pockets, so when they are far away from each other, and pine for the sound of "that beloved voice," they will have only to take up this electrical apparatus, put it to their ears, and be happy. Ah! blissful lovers of the future!