Can a Utah police officer ban a person from a public park even if the person is only suspected of breaking the Law?
Utah has a lot of public parks, so to point in the right direction, I will assume that this is a public park in Salt Lake City, it's just a plain old grassy field, and it's not during a special event. A person is suspected of some crime like selling drugs, not arrested, but told by a police officer to go away and never come back. This is way beyond the power of the police. After due legal process, a proven (not just suspected) public menace could be ordered by the court to stay away from the park.
A police officer can, of course, order a person to leave a park when they violate a park rule, in fact rule number 1 is "It is unlawful for any person to do or to allow or permit any of the acts prohibited by this chapter in any park in Salt Lake City", so the police cannot legally turn a blind eye to rule violations. Violation of park rules is an infraction which can earn you a ticket of up to $299. However, the officer can tell you to go away, rather than giving you a ticket or arresting you. But an police order cannot issue a unilateral restraining order.
Apart from city laws, there are general state laws regarding trespass and destruction of property. The state criminal trespass law says that
A person is guilty of criminal trespass if...knowing the person's... entry or presence is unlawful, the person enters or remains on to which notice against entering is given by...personal communication to the person by the owner or someone with apparent authority to act for the owner
In this case, the owner is the city, and the officer has apparent authority to act for that owner. In the case of private property, the owner or his agent has very broad authority to give notice requiring you to leave (e.g. if you don't like their politics or their shirt); but in the case of public property, the government has more narrowly circumscribed authority to kick you out.
I would think that a person "only suspected of breaking the law" in a public park might be given the choice of leaving or being arrested. It's the kind of question where you'd need to know all the circumstances.
It's also possible that "suspected of breaking the law" is not the only reason, but being "reasonably assumed to be a danger to visitors to the park" might be another one. Or "being suspected of having caused major damage to the park".
Yes, if left unchallenged.
There are people who have been verbally told by public safety agents not to return to governmental properties open to the public (indefinitely) or they will be arrested, even where no law has been alleged or even suspected as being violated; only accusations of certain words being spoken within the facility (that is, lies). (Citation: experience).
The point is all adverse actions by public safety agents which violate the individuals' Constitutional rights need to be immediately documented both privately and by means of the Fourth Estate and challenged administratively and in a court of competent jurisdiction.