And even the tourists' measured coach is willing to concede England may have unearthed "a very, very good player".
"Without wanting to put too much pressure on him, the signs are good," Bayliss said. "For one so young, 19, his composure is very good.
"He knows how his game works - something that players years in advance of him would like to have, I think.
"He's been with the squad for about six weeks, but it feels like he's been here for two or three years - he's got that type of a personality.
"He's one of those batters, a little bit like a Kumar Sangakkara, who just loves batting - whether it's in the nets, having throwdowns or with a tennis ball in the changing room. He just loves to have a bat in his hand."
Hameed's spot seems assured for the foreseeable future, but Bayliss does have selection calls to make ahead of this week's second Test - not least whether to recall James Anderson.
England's leading Test wicket-taker has recovered from a shoulder injury and bowled in the nets in Rajkot, and Bayliss said: "We'll assess him in the next couple of days.
"The medical staff say he's in better shape than he was when he came back and played against Pakistan [last summer]. These last five days have been tough, so we'll have to assess how everyone has come through - and then we'll make a decision."
England have also announced that spin bowling consultant Saqlain Mushtaq will stay with the squad until the end of the third Test in Mohali.