Written and first recorded by Arthur Gunter (1954).
Hit version by Elvis Presley (C&W #5 1955).
Also recorded (as a demo) by Buddy Holly (1955).
From the wiki: “‘Baby Let’s Play House’ was written by Arthur Gunter and recorded by him in 1954. It was covered by Elvis Presley the following year on Sun Records – the fourth issue of a Presley record by Sun … and it became the very first recording by Elvis to appear on a national chart, when it made #5 on the Billboard Country singles chart in July 1955.
“In his youth the songwriter, Gunter, formed the Gunter Brothers Quartet with brothers and cousins. In 1954 Gunter signed with Excello Records and recorded ‘Baby Let’s Play House’ in November 1954. It was released on Excello 2047 and became a local hit. Gunter would later say ‘Elvis got that number and made it famous. But I didn’t get a chance to shake his hand.’ Gunter’s first royalty check, received that same year, was for $6500 (equivalent to $57,000 in 2014).