No, don't do it
While more information is needed, IN GENERAL the answer would be "NO, don't do it!". Think about this: when you create a landing page, you need to create a funnel to make users perform an action (usually registering or purchasing). The funnel image is not random, think on its shape and you'll realize you're going from general to particular, from broad elements to action. Every single element converges on the CTA.
Now, start adding elements. Including your navigation! You'll easily see how you're diverging attention from the only thing that matters to anything else that doesn't matter at all. Try to imagine your users viewing all elements and how their focus will go to several different points in the screen.
The above paragraphs are a simple description of something that has been thoroughly tested: the more distractions, the less CTR (or in other words: the less friction, the better the conversions). Basically, you're adding actions (as in... Call To Action!) so you're offering several CTAs in the same page. Sounds crazy? Well, it is
Your landing page is not your home page
If you're building a landing page, this is because you need to do something different than your homepage. Sometimes, home pages might be landing pages by themselves, but if you need to add different elements (deep navigation, deep linking, search, social media, info and so on), you better build a simple, CTR focused and straight to the point landing page
Not all elements have the same weight
So I mentioned your navigation as well as your search box. It's easy to see that in some cases, you may use navigation for CTR purposes, as long as you don't have a complex navigation. But all in all, it could happen. And at worst, in a navigation you have a more or less clear indication of where are you going, and what will you expect to see once you click.
Now, think on a search box: you're simply adding unpredictability and randomness. Your users won't be clicking a controlled link, but they will look for whatever they want, even things you won't have in your site. As you may imagine, they won't search for "how can I sign up for this amazing site?" but they will look for almost anything you can imagine. Bottom line is: you can't control the user behavior. Instead, you're hoping that after performing a search, they eventually land somewhere useful for you. To better understand this, try reading about locus of control concept.
There might be a few situations where this search box might be useful (for example, if you're looking for an apartment) but most often than not, you should avoid any distraction from what you intend users to do.