The police can arrest you. Once they arrest you, the issue would be whether they could question you and if they did question you, whether the information you revealed would be able to be used against you. That issue would turn on whether you were able to competently waive your right to be silent and have an attorney during questioning.
People think that officers have to read them their rights immediately upon arrest. They do not. Officers only need to read you your rights when they are questioning you and you are in custody.