Yes, it is possible to build a steam powered aeroplane. The first sucessful powered, heavier than air, flying machine was steam powered, built by Henson and Stringfellow in 1847. Several later designers also built sucessful models.
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The electric passenger plane completed its maiden voyage on September 27, 2022, at Moses Lake airport in the US state of Washington. Alice is so far the only passenger aircraft developed from scratch and capable of flying with battery power.