I went abroad for a week to visit family when DD was about that age.
The situation was a bit different in our case though. We didn't know we were having a baby until DP went into labour, and had only been in a relationship for 11 months when she was born. As a result we weren't living together, DP and DD were living with DPs parents. They actually moved in with me the week after I got back.
As a result I'd booked the flights before DD was born, non refundable as I had no idea a baby was on the way. Had I not gone, I'd only likely have seen them for a few hours daily anyway. (Between work and there being no room for me to sleep there, I generally only spent a few hours every evening there). And DP had 3 other people in the house to help her with DD.
In a more normal situation, there's no way I'd have gone away for that long when DD was little. Post them moving in with me, I did have the occasional weekend away with friends from about DD being 5yo, as did DP, but my first full week away from DD and DP was this year, when DD was the grand old age of 18 years old!
I think your partner is taking the piss a bit to be honest OP. It's not something I'd have even considered, let alone actually asked if it was OK