I am trying to fit penalized model for binary outcome with few events and correlated covariates. Probit and logistic regression models are among the most widely used models for binary outcome. I am wondering why penalized logistic regression has been developed and codes written but not penalized probit regression?
Are you sure you really need probit, logit and probit are very close for most uses. The case where they differs is in multivariate models, like multivariate probit for choice modeling, where the multivariate normal underlying the probit is much more flexible for modeling than multivarite logit. Some references for that is Discrete Choice Methods with Simulation which contains much more. There seems to be little on this site, but there is a tag choice and here a stored search.