A play group for a 1 year old is not going to be a drop off experience.
Preschools, of the not full time daycare variety tend to start at either 2 or 2.9. Drop off part time preschool for 2 year olds is usually either 2, 3 or 5 mornings a week.
Most daycares don’t do 1 day a week schedules. Financially it doesn’t make sense for them and they can generally stay fully enrolled. You might find an in home daycare that will take your kid 1 day a week. But, I think that’s most likely if it happens to fit their schedule. They are not going to turn down a full time kid to hold your 1 day a week.