I’m not sure about where you live but where I live (Canada) you can’t legally do your own heat pump installations. You need to be certified to handle refrigerants and that sort of thing.
I'm from the UK. I was under the impression you could do it yourself here but couldn't find a definitive answer.
It's absolutely something I wouldn't take on myself though. Changing a light switch or rewiring a plug, sure - but I'd be guaranteed to make a huge, expensive, inefficient mess with a project like that.
Yes, for example a mini split unit. You can purchase these at the hardware store and install them yourself. You just can’t do this with units that integrate into a central heating/cooling system which require the installation and charging of the refrigerant lines.