The development of both Lie groups and linear algebraic groups is rather complicated, starting with the definitions over various fields. For example, working over $\mathbb{Q}_p$ or a finite extension is one classical setting, and many of the ideas (though not all) carry over well to "local fields" in prime characteristic.

While ACL and Venkatarama have correctly pointed to Serre's lectures for a conventional treatment of Lie groups over local fields (including $\mathbb{R}$ and "ultrametric" fields), it may help to look more broadly at the development of the ideas over time. Chevalley set out to write a six volume series of books on Lie groups (and "linear algebraic groups"), but abandoned that after three books in order to develop an improved theory of linear algebraic groups using a recent version of algebraic geometry. But he had already realized that compact (real) Lie groups carry a natural algebraic group structure (which yields the same Lie algebra), even though most other Lie groups do not.

His student Robert Hooke at Princeton wrote a thesis in 1942, soon published in the Annals here (available through JSTOR). In this work the standard dictionary between Lie groups and Lie algebras is adapted to $p$-adic groups.

In the next decades Bourbaki (no doubt actively influenced by Serre) began to issue chapters of their treatise Groupes et algebres de Lie, later published in English translation by Springer. Chapter II (Hermann, 1972) lays the foundations for a unified theory of Lie groups and their Lie algebras over $\mathbb{R}, \mathbb{C}$, and complete local fields; sometimes the field is required to be of characteristic 0, so one has to be aware of this. It's very useful to consult their historical notes on Chapters I-III.

Already Serre had lectured at Harvard in 1964 on Lie algebras and Lie groups, borrowing some of the not yet published Bourbaki approach. These lectures were published by W.A. Benjamin (1965) and later reissued as Springer Lecture Notes 1500 here.

Though the theory of $p$-adic Lie groups is now well grounded and to some extent unified with traditional Lie group theory, it remains at some distance from linear algebraic groups. However, as in the real and complex Lie group cases, there are much closer connections when the groups are connected and reductive: this has led to a rich literature, including the work of Iwahori-Matsumoto and Bruhat-Tits on structure theory over local fields. By the time Conrad-Gabber-Prasad wrote their recent book Pseudo-reductive Groups, the group scheme approach allowed further refinements in the study of reductive groups, especially in prime characteristic. (But these developments move farther away from the traditional study of manifolds and Lie groups, including the close relationship between Lie groups and their Lie algebras.)

The development of both Lie groups and linear algebraic groups is rather complicated, starting with the definitions over various fields. For example, working over $\mathbb{Q}_p$ or a finite extension is one classical setting, and many of the ideas (though not all) carry over well to "local fields" in prime characteristic.

While ACL and Venkatarama have correctly pointed to Serre's lectures for a conventional treatment of Lie groups over local fields (including $\mathbb{R}$ and "ultrametric" fields), it may help to look more broadly at the development of the ideas over time. Chevalley set out to write a six volume series of books on Lie groups (and "linear algebraic groups"), but abandoned that after three books in order to develop an improved theory of linear algebraic groups using a recent version of algebraic geometry. But he had already realized that compact (real) Lie groups carry a natural algebraic group structure (which yields the same Lie algebra), even though most other Lie groups do not.

His student Robert Hooke at Princeton wrote a thesis in 1942, soon published in the Annals here (available through JSTOR). In this work the standard dictionary between Lie groups and Lie algebras is adapted to $p$-adic groups.

In the next decades Bourbaki (no doubt actively influenced by Serre) began to issue chapters of their treatise Groupes et algebres de Lie, later published in English translation by Springer. Chapter II (Hermann, 1972) lays the foundations for a unified theory of Lie groups and their Lie algebras over $\mathbb{R}, \mathbb{C}$, and complete local fields; sometimes the field is required to be of characteristic 0, so one has to be aware of this. It's very useful to consult their historical notes on Chapters I-III.

Already Serre had lectured at Harvard in 1964 on Lie algebras and Lie groups, borrowing some of the not yet published Bourbaki approach. These lectures were published by W.A. Benjamin (1965) and later reissued as Springer Lecture Notes 1500 here.

Though the theory of $p$-adic Lie groups is now well grounded and to some extent unified with traditional Lie group theory, it remains at some distance from linear algebraic groups. However, as in the real and complex Lie group cases, there are much closer connections when the groups are connected and reductive: this has led to a rich literature, including the work of Iwahori-Matsumoto and Bruhat-Tits on structure theory over local fields. By the time Conrad-Gabber-Prasad wrote their recent book Pseudo-reductive Groups, the group scheme approach allowed further refinements in the study of reductive groups, especially in prime characteristic. (But these developments move farther away from the traditional study of manifolds and Lie groups, including the close relationship between Lie groups and their Lie algebras.)